Estimating the design flood under the influence of check dams by removing nonstationarity from the flood peak discharge series

The construction of check dams in northwestern China has resulted in nonstationary changes in flood peak discharge series; the stationary assumption of the conventional hydrological frequency analysis is no longer satisfied. According to the characteristics of the construction and operation of check dams, the nonstationarity of flood peak discharge series are largely induced by changes in the effective runoff generation area (i.e., the basin area minus the area controlled by check dams). Knowing the power function relationship between the flood peak discharge and the basin area, we can remove the influence of the effective runoff generation area and convert the original nonstationary series into a stationary series. This de-nonstationarity method can achieve stationarity in the first and second moments simultaneously. Therefore, we can calculate the design value of the reconstructed series using the conventional frequency analysis method. According to the effective runoff generation area under design conditions, we can then obtain the corresponding design flood of the original series. We applied this method to the Mahuyu River basin to obtain the design flood under nonstationarity. Due to the consideration of the deterministic influence of check dams during the de-nonstationarity process, the uncertainty analyzed by the bootstrap method is obviously small.


INTRODUCTION
Design flood estimation plays an important role in water project design, water resources planning, and flood risk control. It is usually obtained using frequency analysis which is a technique of fitting a probability distribution to a series of observations for defining the probabilities of future occurrences of some events of interest, e.g., an estimate of a flood magnitude corresponding to a chosen risk of failure (Khaliq et al. ). The conventional hydrologic frequency analysis method requires the hydrological series to be independent and identically distributed (Salas & Obeysekera ; Qin et al. ; Read & Vogel ). For the flood peak discharge Q with distribution F Q (Qjθ), we generally assume that the series is stationary, and that the distribution parameters θ are time invariant. In northwestern China, climate and topographical conditions cause serious water and soil loss.
As such, many check dams have been built to try to mitigate this phenomenon. In the context of the construction of thousands of check dams, the stationarity of the flood peak discharge series gradually disappears. The nonstationary flood frequency analysis has become a point of concern for engineers and local managers.
Under nonstationary conditions, in the absence of understanding the physical mechanisms, we usually assume that the distribution parameters of the hydrological series change with time; the time-varying distribution is determined by fitting the time-varying characteristic of the distribution parameters from a statistical perspective.
Based on this logic, Strupczewski et al. (a, b) and Strupczewski & Kaczmarek ()  In order to cope with nonstationarity, some studies have extended the concept of return period to a more general interpretation as the expected waiting time (EWT) Even if meteorological or anthropogenic factors are chosen as the covariates, the nonstationary changes are usually characterized using data-based regression technique. This nonstationary change is usually fitted with linear or some simple nonlinear curves, but the actual conditions must be extremely complicated, and it is difficult to answer how this nonstationary change will evolve in the future. Besides, according to statistical theory, because the nonstationary hydrological series no longer has ergodicity, the population distribution cannot be inferred from it.
For a long time, we have tried to reformulate and extend the conventional hydrological frequency analysis method to adapt the nonstationary conditions. Since there is insufficient knowledge of the physical mechanism, for nonstationary frequency analysis, new challenges and doubts are encountered as one problem after another is solved.
However, under the stationary conditions, the conventional frequency analysis method has been widely proved to be scientific and reasonable. The conventional method has been mastered by engineers, and there are a large number of guides and criteria to ensure its reliability. Therefore, the conventional frequency analysis method should not be discarded easily. If the stationarity of a hydrological series can be reconstructed, then the conventional method can still be used to calculate the design value.

METHODOLOGY
Removing nonstationarity from the flood peak discharge series The presence of the check dams resulted in a reduced effective runoff generation area, which leads to nonstationary changes in the flood peak discharge series. After removing the influence of the effective runoff generation area from the original nonstationary series, the remaining part of the flood peak discharge series should be statistically stationary.
Numerous studies have shown that there is a power function relationship between the flood peak discharge and the basin area (Smith ; Gupta ). This power function relationship in the Yulin region takes the form of (Quan ): where A is the basin area. A more complete expression is as follows: the influence of all these factors as ξ, and assume that it is a stationary random variable with a probability distribution characterized by the mean μ and the variance σ 2 . By rearranging Equation (2), we can remove the nonstationary influence of the effective runoff generation area from the flood peak discharge series. This reconstructed series RS(t) is defined as follows: The mean and variance of the reconstructed series RS(t) are as follows:  (2), the design value for the original nonstationary flood peak discharge series is as follows: where rs p is the design value of the reconstructed series Serinaldi & Kilsby () pointed out that the bootstrap method strictly depends on the available information without any asymptotic assumptions, and that it does not rely on specific parameter estimation methods. Moreover, this method is relatively easy to implement, regardless of the complexity of the model. We use the bootstrap method to estimate the confidence intervals of the design flood. The procedure of the bootstrap method is as follows:

RS(t) and
1. After removing the nonstationarity from the original flood peak discharge series Q(t), we apply the conventional frequency analysis method to stationary reconstructed series RS(t) and calculate the design value rs p for a certain return period.
2. The design value Q p of the original nonstationary flood peak discharge series is calculated with Equation (6).
3. We then resample the reconstructed series RS(t) with replacements to obtain a new sample series RS Ã (t).
4. Based on the new sample series RS Ã (t), we calculate the new design value rs Ã p for a specific return period. 5. Using Equation (6), we determine the design value Q Ã p of the nonstationary flood peak discharge series for a specific return period.

Stationary analysis of reconstructed series
After removing the nonstationary influence of the effective runoff generation area, we obtained the stationary reconstructed series RS(t) (Figure 4). The hypothesis test results show that both the first and second moments of the reconstructed series of the MRB flood peak discharge are stationary (Table 2) (Table 3).
In order to select the best-fitting distribution, we evaluated the overall simulation accuracy of the alternative   Table 4.
For stationary series RS(t), the K-S test statistic and NSE pp show that the GEV distribution is optimal, while NSE QQ and RMSE indicate that the WEI distribution is a better fit. In general, the difference between the empirical probability and the theoretical probability for the same sample point at the tail of the distribution is small (both close to 1), but the difference between the empirical and theoretical quantile is relatively larger. Therefore, NSE QQ is more sensitive to the tail than NSE pp . We conclude that the WEI distribution is the optimal distribution for the reconstructed series of the MRB flood peak discharge. The Q À Q plot in Figure 5 also shows the good performance of the optimal distribution.

Calculation of the design quantiles
We estimated the distribution parameters using the Lmoments method (Hosking ). The L-moment is a linear combination of the probability weighted moments.
Compared with the regular moment estimation method, the L-moments method is less affected by individual data points. After removing the influence of the effective runoff generation area from the flood peak discharge series, we estimated the distribution parameters of the reconstructed stationary series (Table 5) and calculated the design quantiles for a certain return period ( Figure 6).   peak discharge is 1,829 m 3 /s, which is generated within the effective runoff generation area of only 56.5 km 2 . This apparently larger, even somewhat impractical, design flood is unreliable. Because of the nonstationarity of the flood peak discharge series, it is obviously impossible to accurately assess the risk of flood by using conventional methods. Besides, the uncertainty of the design flood calculated by the original nonostationary flood peak discharge series is larger than that calculated by the de-nonstationary method. When the return period is more than 50 years, the width of uncertainty CIs of the design value by the conventional method has exceeded the design value itself.
According to the physical mechanism of check dams on flood peak discharge, the de-nonstationarity method    Table 7 (other details are not shown).
As shown in Table 7, the design flood calculated by the DLL method is the largest, followed by the ENE method.
The design values of these two methods even exceed that of the conventional method, which is obviously unreasonable. The design flood of the ENE method increased rapidly with the increase of the return period, and almost did not change when the return period was more than 20 years. This change characteristic of the design value of the ENE method is also unrealistic. The design value of ER and ADLL methods are relatively small and close to that of the de-nonstationarity method. However, the uncertainty of these two methods increases rapidly as the recurrence period increases, and even exceeds the magnitude of the design value. On the other hand, these methods all take the effective runoff generation area as the covariate to consider its influence. In MRB, although the effective runoff generation area of a flood event is significantly reduced, the runoff generation process outside of the control area of check dams should not change, that is, the flood peak modulus should not change. The flood peak modulus of the design flood obtained by these reliability-based methods and the conventional method is obviously increased. The denonstationarity method proposed in this paper has certain advantages in terms of the magnitude of the calculated design value, the uncertainty, and the consideration of the physical mechanism.  The conventional method, which requires hydrological series to be stationary, is a mature and robust scientific strategy. In the presence of check dams, nonstationarity in the flood peak discharge series is caused by changes in the effective runoff generation area. By quantifying the power function relationship between the flood peak discharge and the basin area, we can remove the nonstationarity from the flood peak discharge series. Once the reconstructed series is stationary, we can calculate the design quantile via the conventional method, and then obtain the design value of the original nonstationary series according to the effective runoff generation area under design conditions.
We applied this de-nonstationarity method to the nonstationary MRB peak flood discharge series. After removing the nonstationary influence of the effective runoff generation area, the nonstationary flood peak discharge series was transformed into the stationary reconstructed series, and the stationarity is simultaneously achieved in the first and second moments. Because the nonstationarity in the effective runoff generation area has a stronger impact on the second moment of the flood peak discharge series than it does on the first moment, we cannot ignore nonstationarity in higher order moments in a changing hydrological environment. The design flood calculated using the de-nonstationarity method is obviously smaller than that calculated according to the original nonstationary flood peak discharge series. By removing the nonstationarity from the flood peak discharge series, this method can greatly reduce the uncertainty of the design flood. Compared to the four reliability-based methods, the de-nonstationarity method has certain advantages in terms of the magnitude of the calculated design value, the uncertainty, and the consideration of the physical mechanism.
The nonstationary problems caused by climate change or human activities cannot be solved through statistical approaches alone. It is necessary to explore the physical mechanism of the nonstationary change and establish a new method of combining physical mechanism with statistical theory. For other cases, it should also be possible to eliminate the nonstationary effects of influence factors on the hydrological series according to the influence law.
Therefore, the exploration of the physical mechanism of the interaction between hydrological elements should be the focus of future research, which can also promote our understanding of hydrological processes.