Theoretical derivation for the exceedance probability of corresponding flood volume of the equivalent frequency regional composition method in hydrology

The ability to estimate the design flood in a given return period is a fundamental issue in engineering design, as well as water resource management and planning. Hydrological frequency analyses have been used worldwide as a standard approach for estimating design flood (Kendall & Stuart 1979; Ponce 1989; Hu et al. 2018). The estimation of design flood generally of interest for hydrologists, engineers, and agriculturalists for the design of hydraulic structures, such as river sections and dam sites. When control engineering has not been implemented upstream of a future dam site, the design flood of the future dam can be directly calculated via a hydrological frequency analysis of the peak or duration of the flood volume series, and the flood regional composition does not need to be considered (Maidment 1992). However, for cases in which one or more control engineering features have been implemented upstream of the future dam site, such as a cascade reservoir system, the impact of the outflows at upstream site and intermediate catchment (i.e., the flood regional composition) must be considered to estimate the design flood at the future dam site. Generally, the framework for flood regional composition (Lu et al. 2012; Guo et al. 2018) consists of a first step, where a proper combination of floods at various sub-catchments is researched in natural conditions, and a second step, where the design flood (flood discharges or volumes) at the downstream site under the influence of upstream reservoir is obtained through flood routing by incorporating the reservoir operation rules. The first step of regional composition assumes that no reservoirs exist and that each sub-catchment is in a natural condition. When a design flood event with exceedance probability p is selected from the natural flood magnitude–frequency curve at a downstream site, an appropriate combination of floods is required that occurred at upstream sites. The corresponding natural design flood hydrograph at sub-catchments of upstream sites is derived by using the same flood amplification ratio of their respective typical flood hydrograph (Yue et al. 2002; Xiao et al. 2009). Before the characteristics of the reservoir are determined, the most important step is to find an appropriate combination of floods that occurred at upstream sites (Boughton & Droop 2003).

For on-site hydrological frequency analyses that do not consider the impact of flood regional composition, the process includes the selection of distribution functions and the estimation of parameters (Rao & Hamed 2000; Badreldin et al. 2012; Zeng et al. 2012). Based on the sample series, a design flood with a given return period is easily estimated (Kirby 1974). In different countries or regions, the recommended distribution functions used to fit extreme flood series may vary; for example, the Pearson type III (PE3) distribution has been implemented in China, the log PE3 distribution has been implemented in the United States, and the generalized extreme value (GEV) distribution has been implemented in England. In the parameter estimation of a distribution function, the method of moments (MOM) was once a widely used approach, although it has a high degree of bias (Wallis et al. 1974; Greenwood 1979). Thus, other methods with less bias have been successively provided, such as the maximum-likelihood method (MLM), probability weighted moments method (PWM), weight function method, and linear moments method (Hosking 1986, 1990; Ding et al. 1989; Liang et al. 2014; Wang et al. 2015). Due to the impact of climate change and anthropogenic activities, many studies from the past several decades indicate that the non-stationary nature of hydrological extreme series has become increasingly significant, and the non-stationary hydrological frequency analysis method has drawn significant attention from researchers and engineering practitioners (Xiong et al. 2015; Hu et al. 2017, 2018).

Compared with on-site hydrological frequency analysis, a multi-site hydrological frequency analysis that considers the flood regional composition is more complicated and difficult. This process should consider not only the design flood of each site but also the influence of flood regional composition on the design flood (flood discharges or volumes) of the future dam site. It can be seen that the number of possible regional composition is countless and the selection of an appropriate combination is significant. In practice, several combinations, such as the best, the worst, and the most likely, are used to simulate the impact of upstream sites on design flood at downstream site (Nijssen et al. 2009). With respect to the design flood regional composition analysis, semi-theoretical and semi-empirical methods, such as the regional composition method, frequency combination method, and stochastic simulation method, have been widely practiced for decades. The regional composition method specifies that a flood occurs in one catchment with the same exceedance probability as in the design section, and a corresponding flood occurs in the other catchments (Ministry of Water Resources 2006). Among various possible compositions, the equivalent frequency regional composition (EFRC) method is able to select a specific one as the designed regional composition model to ensure the safety of the calculated results (Lu et al. 2012; Guo et al. 2018). The EFRC method includes two cases. In the first case, the corresponding flood at intermediate catchment is calculated if the exceedance probabilities of the design flood volume at upstream site and downstream site are equivalent. In the second case, the corresponding flood at upstream site is calculated if the exceedance probabilities of design flood volume at intermediate catchment and downstream site are equivalent. The EFRC method is a back-calculation model based on the water balance, which is used to calculate the corresponding flood volume for an intermediate catchment without stream gauging stations. It plays an important role in multi-site hydrological frequency analysis, especially in China (Liang et al. 2016).

Regardless of which flood regional composition method is used, the volume of the corresponding flood is known, while the exceedance probability (or return period) of the corresponding flood volume is unknown. Using one case of the EFRC method as an example, such as one in which the equivalent exceedance probability of the design flood volume at upstream site and downstream site is used to calculate the corresponding flood at intermediate catchment, the corresponding flood volume is equivalent to the gap between the volumes of upstream and downstream floods. Whether the exceedance probability of the corresponding flood volume is larger or smaller than that of the design flood volume at downstream site has not been theoretically proven, which has introduced confusion into practical probabilistic applications.

The aim of our study is to investigate this unresolved question and to clarify it using theoretical derivations and Monte Carlo (MC) experiments. Considering that different countries or regions use various flood distributions, we selected six representative design flood volume distributions for our research. The normal, extreme value type I (EV1(2)) and logistic distributions use theoretical derivations because the joint distribution functions of them are easy to derive. The PE3, two-parameter lognormal (LN(2)) and GEV distributions are used to carry out MC experiments. Since their joint distribution functions are transcendental functions, it is difficult to calculate them directly.

The flood regional composition is used to determine the design flood volumes at downstream sites. Many possible flood regional combinations occur in various sub-catchments. Different combinations can result in different design flood volumes at downstream site.

The cascade intermediate catchments system, as shown in Figure 1(c), can be divided into many small single intermediate sub-catchments, as shown in Figure 1(b). Every single sub-catchment has a similar design flood volume composition. As the number of sub-catchments (n) increases, the number of flood regional compositions (n) increases uniformly.

The EFRC method is widely used to determine the design flood regional composition in China, and it is recommended by the Ministry of Water Resources of China (Ministry of Water Resources 2006; Guo et al. 2018). When there is no significant overstandard in the measured data, the design flood regional composition can be deduced by using this method. To describe the EFRC method, we consider a single intermediate sub-catchment, as shown in Figure 1(b), with one upstream site, an intermediate catchment, and a downstream site. The inflow of downstream site Z contains the outflow at upstream site X and the runoff at intermediate catchment Y. The EFRC method assumes that the exceedance probability of the sub-catchments (upstream site X or intermediate catchment Y) is equivalent to that of downstream site Z, whereas the flood volume at the other sub-catchments is obtained by back-calculation with respect to the water balance. In engineering practice, the EFRC method has two forms, as follows.

We take the first form of the EFRC method as an example and determine whether P > C or P ≤ C via theoretical derivations and MC experiments in the following.

In this study, the normal, EV1(2) and logistic distributions are used to investigate the relationship between the exceedance probability C of corresponding flood volume at intermediate catchment and exceedance probability P of design floods at upstream site and downstream site via theoretical derivations. For the PE3, LN(2) and GEV distributions, which are transcendental functions that are difficult to calculate, the relation between C and P is investigated via MC experiments, as shown in the following sections, and then extended to the general distribution.

According to Equation (8), the exceedance probability of flood is equal to 42.96%.

According to Equation (12), the exceedance probability of flood is equal to 50%.

Assume that the flood volume at upstream site, intermediate catchment, and downstream site is subject to normal distributions of ⁠, ⁠, and ⁠, independently.

Let ⁠, and ⁠, ⁠. As shown in Figure 1(b), the downstream flood volume Z equals the sum of upstream flood volume X and intermediate flood volume Y. Thus, and ⁠, which means that the distribution parameters of the random variables Y and Z can be represented by the statistics of X, ⁠, and ⁠.

⁠, ⁠,

where and ⁠.

1. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is larger than C.

2. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P equals C.

3. If ⁠, i.e., ⁠,

⁠, ⁠,

then P is less than C.

Overall, in the context of using the first EFRC method for normal distribution floods, the relationship between C and P depends on whether or not P is larger than 50%. Thus, for a design flood (volume) whose exceedance probability is generally less than 50%, C is greater than P; however, for low-flow condition whose exceedance probability is generally larger than 50%, C is less than P. For example, if the return periods of the volumes of upstream and downstream floods are both 1,000 years, then the return period of intermediate flood volume could be 100 years, and if the guarantee rates of upstream and downstream flows are both 90%, then the guarantee rate of intermediate flow volume could be 70%.

In the context of other EFRC methods, similar conclusions can be drawn for a normal distribution flood.

Let ⁠,

⁠, ⁠,

where and

1. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is larger than C.

2. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P equals C.

3. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is less than C.

Thus, when using the first EFRC method for the EV1(2) distribution flood, the relationship between C and P depends on whether P is larger than 42.96%. For a design flood (volume) whose exceedance probability is generally less than 42.96%, C is larger than P, whereas for the design low flow whose exceedance probability is generally larger than 42.96%, C is less than P.

⁠, ⁠,

where and

1. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is larger than C.

2. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is equal to C.

3. If ⁠, i.e., ⁠,

   ⁠, ⁠,

   then P is less than C.

When using the first EFRC method for flood regional composition with logistic distributions, the relationship between C and P depends on whether P is larger than 50% or not. Thus, for a design flood (volume), C is greater than P, whereas for a design low flow, C is less than P.

For a PE3 distribution as shown in Equation (13), the multivariate PDF is a transcendental equation. Therefore, a theoretical form of derivation for the relationship between P and C may not be obtained. Then, MC experiments are used to infer whether P > C or P ≤ C.

MC experiments are performed to produce and utilize random numbers to solve complex calculations (Denny & Yevjevich 1972; Christiane 2009; Xing et al. 2019). Specifically, for the problem to be solved, a random variable is constructed, a large number of random numbers are sampled according to the variable's numerical characteristics, and the corresponding parameter values are calculated from these samples as the solution to the problem. Figure 2 shows a flow chart that clearly illustrates the procedure of MC experiment. In the figure, the steps of MC experiment are as follows.

Step 1: Randomly generate upstream flood volume Xrng and downstream flood volume Zrng according to their numerical characteristics (EX, EZ, Cs, and Cv). Zrng minus Xrng is internal flood volume Y.

Step 2: Repeat Step 1 100,000 times. Obtain 100,000 random internal floods Y at the same time.

Step 3: Plot the hydrologic frequency analysis curve of Y through 100,000 random numbers.

Step 4: Plot hydrologic frequency analysis curves of Z and X through their numerical characteristics.

Step 5: Based on design probability P, design floods Zp and Xp are obtained from the hydrologic frequency analysis curves of Z and X, respectively.

Step 6: Zp minus Xp is the corresponding flood volume Yc.

Step 7: Based on the Yc, exceedance probability C is obtained from the hydrological frequency analysis curve of Y.

Step 8: Repeat Steps 5–7 for design probability P of 0.01, 0.1, 1, 2, 10, 20, 25, 30, 35, 40, 45, 50, 60, 75, 80, 90, 95, 97, 99, and 99.9%. Compare the relationship between P and C. Find critical point P₀.

For the PE3 distribution, we performed 100,000 random trials for the flood regional composition. In each of these trials, we produced the random upstream flood volume X and downstream flood volume Z according to their numerical characteristics (⁠⁠, ⁠, and ⁠), and we identified the corresponding flood volume at intermediate catchment Y based on the difference between the volumes of downstream and upstream floods. Then, we obtained the cumulative distribution of Y through 100,000 corresponding intermediate floods volumes. Considering the universality and extensive suitability of the EFRC method, six regional composition schemes were constructed as representatives. To ensure that the downstream flood volume was always greater than the upstream flood volume, the of upstream site was 1,000 and the of downstream site was 2,000. For and ⁠, both upstream and downstream presented values of ⁠, ⁠, ⁠, ⁠, ⁠, and ⁠. Therefore, six regional composition schemes are obtained in total. Design probabilities P of each regional composition scheme are 0.01, 0.1, 1, 2, 10, 20, 25, 30, 35, 40, 45, 50, 60, 75, 80, 90, 95, 97, 99, and 99.9%.

A specific regional composition scheme is used as an example, such as when the parameters of downstream floods (volume) are = 2,000, = 0.4, and = 4 and those of upstream floods (volume) are = 1,000, = 0.4, and = 4. The results of the statistical experiments are shown in Table 1, and the relationship between the exceedance probabilities of C and P is presented in Figure 3 (the same as the regional composition scheme 2 in Figure 4).

As illustrated in Figure 3, a critical point P₀ is presented, which is the exceedance probability corresponding to the intersection of two lines. When the exceedance probability of design flood P is greater than the critical point P₀, then the exceedance probability of corresponding flood at intermediate catchment C is less than P (C < P). When P is exactly equal to P₀, then C = P. When P is smaller than P₀, then C > P. All of these results show that in the context of using the first EFRC method for PE3 distribution, the relationship between C and P depends on the design probabilities of the volumes of upstream and downstream floods. This conclusion is consistent with that drawn from the theoretical derivation for normal, EV1(2) and logistic distributions. More specifically, the P₀ of normal, EV1(2) and logistic distributions are 50, 42.96, and 50%, respectively, whereas the P₀ of PE3 distribution is not fixed and may be greater than, less than, or equal to 50%.

P₀ of the typical regional composition in Figure 3 is 45%, which is less than 50%. Similarly, P₀ of the other five regional compositions could be obtained. All of the results are shown in Figure 4 and Table 2, the maximum of P₀ is 50% and the minimum of P₀ is 30%; namely, the critical point P₀ of PE3 distribution is between 30 and 50%. In addition, because of the various values in the experiments, the range of P₀ of PE3 distribution is broad (from 30 to 50%). When the is larger than 2.0, the curve of the PDF of PE3 distribution looks like an ‘L’ shape, and when is less than 2.0, the curve is shaped like a bell. The ‘L’ shape means that the value of a variable has the greatest likelihood near its minimum value, which does not conform to the hydrological phenomenon. For the hydrological variables, the chances of occurrence of extremely large value and extremely small values are very low, while the chances of occurrence of intermediate values are higher; that is, the curve should be shaped like a bell. Therefore, the PE3 distribution of > 2.0 is not suitable in hydrology. However, in most cases, the value of is not restricted in practice. Therefore, we constructed six regional composition schemes with six different values, including > 2.0. If we do not consider the case of > 2.0, the range of critical point P₀ of the PE3 distribution will be between 45 and 50% (as shown in Table 2).

Taking into account the actual situation of design flood regional composition, the design probabilities of the volumes of upstream and downstream floods are generally small, such as P = 0.01, 0.1, or 1%; and these values are far less than 30%. Therefore, the exceedance probability of corresponding intermediate flood volume is greater than that of the volumes of upstream and downstream floods. For example, if a 100-year design flood (P = 1%) occurs in an upstream site and a downstream site, then the corresponding flood volume at intermediate catchment will be less than the 100-year intermediate flood volume. For low-flow regional compositions, the design exceedance probability (or design guarantee rate) of upstream and downstream design flows is generally greater than 50%, such as P = 75, 90, or 99%. Therefore, the exceedance probability of corresponding intermediate flow volume is less than that of upstream and downstream flows. For example, if a 90% guarantee-rate design flow occurs in upstream site and downstream site, then the guarantee rate of corresponding flow at intermediate catchment will be less than 90%.

For the flood regional composition of LN(2) distribution (Equation (15)), we performed 100,000 randomized trials, which is similar to the experiment analysis of the PE3 distribution. Six regional composition schemes were constructed as representatives. For the downstream site, is 2,000, and for upstream site, is 1,000. The values of both upstream and downstream floods (volume) are 0.2, 0.4, 0.5, 0.7, 1.0, and 1.5.

We then used a specific regional composition scheme as an example, such as one in which the parameters of downstream flood (volume) are = 2,000 and = 0.7, and the parameters of upstream flood (volume) are = 1,000 and = 0.7. The results of the statistical experiment and the relationship between the exceedance probabilities of C and P are shown in Figure 5 (the regional composition scheme 4). For this scheme, the critical point P₀ of the LN(2) distribution is 45%, which is less than 50%.

Similarly, the critical point P₀ of the other five regional compositions could be obtained. All of the results are shown in Figure 5 and Table 2. The critical point P₀ of LN(2) distribution is between 45 and 50%.

Therefore, for the design flood regional composition of LN(2) distribution, the exceedance probability of corresponding intermediate flood volume is greater than that of the volumes of upstream and downstream floods. For low-flow regional composition, the exceedance probabilities of corresponding intermediate flow volume are less than the design probabilities (or design guarantee rates) of upstream and downstream flows.

For GEV distribution (Equation (18)), we also performed 100,000 randomized trials. In the six regional composition schemes, the of downstream site was 2,000 and that of upstream site was 1,000. For the and ⁠, both upstream and downstream presented values of ⁠, ⁠, ⁠, ⁠, ⁠, and ⁠.

A specific regional composition scheme is used as an example, such as one in which the parameters of downstream floods (volume) are = 2,000, = 0.8, and = 0.2, and the parameters of upstream floods (volume) are = 1,000, = 0.8, and = 0.2. The results of the statistical experiment and the relationship between the exceedance probabilities of C and P are shown in Figure 6 (the regional composition scheme 1). For this scheme, the critical point P₀ of a typical GEV distribution is approximately 50%, which is similar to the values of the PE3 and LN(2) distributions.

In the same way, the critical point P₀ of the other five regional compositions could be obtained. All of the results are shown in Figure 6 and Table 2, the critical point P₀ of the GEV distribution is between 45 and 50%.

In the actual design flood regional composition, the intermediate exceedance probability is greater than the design exceedance probability. For the design low-flow regional composition, the exceedance probability is less than the design exceedance probability (or design guarantee rate).

Table 3 presents all the results of derivation and experiment analysis. In summary, for the EFRC method with different flood distribution types, the relationship between the intermediate exceedance probability C and design exceedance probability P can be determined through theoretical derivations or statistical experiments to achieve consistent conclusions: For design flood regional composition, the exceedance probability of the corresponding intermediate flood volume is greater than the design probabilities of the volumes of upstream and downstream floods. For example, if a 100-year design flood (volume) occurs in upstream site and downstream site, then the return period of corresponding flood volume at intermediate catchment is less than 100 years. For low-flow conditions, the exceedance probability of the corresponding intermediate flow volume is less than the design probabilities of upstream and downstream flows. Thus, if a 90% guarantee-rate design flood (volume) occurs in upstream site and downstream site, the guarantee rate of corresponding flood volume at intermediate catchment is less than 90%.

In the EFRC method, which is commonly used to resolve the regional composition of the design flood volume at various sub-catchments in natural conditions, the exceedance probability (or return period) of the estimated corresponding flood volume is unknown. This study performs theoretical derivations and MC experiments to investigate the relationship between the probability of the corresponding flood volume and that of the design flood volume at the dam site. The following conclusions are obtained.

1. Critical probability value P₀ exists in the EFRC method. When the exceedance probability of downstream design flood volume P is greater than P₀, the exceedance probability of corresponding flood volume C is less than P, i.e., (C < P); however, when P is equal to P₀, C = P, and when P is less than P₀, C > P. The value of P₀ is related to the distribution function of hydrological variables. For normal distribution, EV1(2) distribution and logistic distribution, P₀ equals 50, 42.96, and 50%, respectively. For PE3 distribution, LN(2) distribution and GEV distribution, P₀ is not fixed. For the PE3 distribution, P₀ ranges from approximately 30–50%, whereas for both LN (2) and GEV distributions, P₀ ranges from approximately 45 to 50% based on the MC experiments.

2. In terms of a design flood event, the design exceedance probability P is generally far less than 30% (e.g., P = 0.01, 0.1, or 1%); thus, the corresponding exceedance probability C is greater than P. For example, if 100-year design floods (P = 1%, i.e., return period T = 100) occur in upstream site and downstream site, the return period of corresponding flood volume at intermediate catchment is less than that of a 100-year event. Therefore, in practical application, the upstream and downstream catchments are very similar in terms of response time, while the intermediate catchment is very small compared to the others, that is the intermediate catchment volume is not as ‘critical’ as those of the other sub-catchments for large T (return period). Besides, when the flood control target at downstream site has been set, the flood control standard of intermediate catchment or upstream site is not as high as that of downstream site, which leads to the prevention of the excessive capacity of flood diversion and peak cutting at upstream reservoirs.

3. In terms of a design low-flow event, the design exceedance probability (or design guarantee rate) P is generally greater than 60% (e.g., P = 75, 90, or 99%); thus, the corresponding exceedance probability C is less than P. For example, if 90% guarantee-rate design flows occur in upstream site and downstream site, the guarantee rate of corresponding low-flow at intermediate catchment is less than 90%, meaning a lower guarantee-rate standard at intermediate catchment.

4. The above conclusions also apply to other regional composition analyses with the EFRC method. For example, if the exceedance probability of design floods at intermediate catchment and downstream site are equivalent, then the corresponding flood volume at upstream site can be calculated. In addition, for other statistical distribution models, the critical probability P₀ would be different, and the corresponding conclusions would be different, although the method provided in this study can be used for reference.

The data in this study were randomly generated through statistical experiments.

This study was supported by the Key Special Project of the National Key Research and Development Program of China (2016YFC0402709, 2016YFC0402706), the Major Program of the National Natural Science Foundation of China (41730750), and the National Natural Science Foundation of China (51709073). The authors extend their sincere thanks to all who were involved in this paper.

All relevant data are included in the paper or its Supplementary Information.