A complete flood event or a flood hydrograph contains multiple feature variables, such as flood peak and flood volume, which can be associated with the safety of hydraulic structures (Salvadori et al., 2004, 2007, 2011; Xiao et al., 2009; Xiong et al., 2015; Loveridge et al., 2017; Shafaei et al., 2017). For example, the water level of a reservoir is controlled not only by flood peak flow but also by flood volume (Salvadori et al., 2011). Therefore, multivariate hydrologic design, which takes into account multiple flood characteristics as well as their dependence, provides a more rational design strategy for hydraulic structures compared to univariate hydrologic design (Zheng et al., 2013, 2014; Balistrocchi and Bacchi, 2017).

Multivariate hydrologic design under stationary conditions has been widely investigated, and the design criterion is usually quantified by the return period, similar to univariate hydrologic design. Under the definition of the average recurrence interval between flood events equaling or exceeding a given threshold (Chow, 1964), the return period of a given flood event under stationary conditions theoretically equals the average inter-arrival time between flood events divided by the exceedance probability (Salvadori et al., 2011). On the other hand, the exceedance probability of a univariate flood event is usually uniquely defined without ambiguity, whereas the exceedance probability of a multivariate flood event could have multiple definitions (Salvadori and De Michele, 2004; Salvadori et al., 2011; Vandenberghe et al., 2011). To date, at least five kinds of different exceedance probabilities for a multivariate flood event have been defined: (1) the OR case in which at least one of the flood features exceeds the prescribed threshold, (2) the AND case in which all flood features exceed the prescribed thresholds, (3) the Kendall case in which the univariate representation transformed from Kendall's distribution function exceeds the prescribed threshold, (4) the survival Kendall case in which the univariate representation transformed from survival Kendall's distribution function exceeds the prescribed threshold, and (5) the structural case in which the univariate representation transformed from a structure function exceeds the prescribed threshold (Favre et al., 2004; Salvadori and De Michele, 2004, 2010; Salvadori et al., 2007, 2013, 2015, 2016; Vandenberghe et al., 2011; Requena et al., 2013; Zheng et al., 2014).

Due to climate change as well as certain anthropogenic driving forces (Milly et al., 2008), the nonstationarities of both univariate and multivariate flood series have been widely reported (Xiong and Guo, 2004; Villarini et al., 2009; Vogel et al., 2011; López and Francés, 2013; Bender et al., 2014; Xiong et al., 2015; Blöschl et al., 2017; Kundzewicz et al., 2018). The multivariate flood distribution exhibits more complex nonstationarity behaviours than the univariate distribution, including nonstationarities of individual margins and the dependence structure between the margins (Quessy et al., 2013; Bender et al., 2014; Xiong et al., 2015; Kwon et al., 2016; Sarhadi et al., 2016; Qi and Liu, 2017; Vezzoli et al., 2017; Bracken et al., 2018; Salvadori et al., 2018). Both nonstationarities of the margins and dependence structure could impact the multivariate hydrologic design. Under nonstationary conditions, the exceedance probability p of a given flood event varies from year to year; thus, the return period, calculated as the average inter-arrival time between two successive flood events divided by p, is no longer a constant (Salas and Obeysekera, 2014; Jiang et al., 2015a; Kwon et al., 2016; Sarhadi et al., 2016; Yan et al., 2017). As a result, a given flood event would correspond to a time-varying and non-unique return period. Consequently, the traditional return-period-based method for estimating hydrologic design may no longer be applicable to engineering practice under nonstationary conditions (Salas and Obeysekera, 2014).

Although increasing attention has been focused on the hydrologic designs under nonstationary conditions in recent years, the focus has mainly been on univariate designs (Obeysekera and Salas, 2014, 2016; Read and Vogel, 2016). To overcome the limitation of the traditional return period under nonstationary conditions, the concept of the return period has been revisited. Salas and Obeysekera (2014) extended two concepts of the return period into a nonstationary framework, defined as the expected waiting time (EWT) for an exceedance to occur (Olsen et al., 1998), and the time period that results in the expected number of exceedances (ENE) equal to 1 over this period (Parey et al., 2010).

Risk and reliability are both important measurements for assessing hydrologic designs (Vogel, 1987; Read and Vogel, 2015). Besides redefinitions of the return period, some risk-based or reliability-based metrics have been introduced as the hydrologic design criteria under nonstationary conditions (Rosner et al., 2014). Rootzén and Katz (2013) proposed the concept of the design life level (DLL) to quantify hydrologic risk in a nonstationary climate during the entire design life period of hydraulic structures. Read and Vogel (2015) introduced the concept of average annual reliability (AAR) to estimate the hydrologic designs under nonstationary conditions. Liang et al. (2016) defined the equivalent reliability (ER) to estimate the design flood under nonstationary conditions by linking the DLL to the return period. Salvadori et al. (2018) associated hydrologic designs with both given life times and failure probabilities to calculate bivariate design values under nonstationarity. These design criteria assess the risk or reliability of hydraulic structures associated with the flood distribution during the entire design life period, rather than for a single year. For a given design life period, these criteria can always yield a unique risk or reliability; therefore, they are applicable to the hydrologic designs under both stationary and nonstationary conditions.

Under the multivariate framework, a given design level would correspond to an infinite number of possible hydrologic design events (Hawkes, 2008; Kew et al., 2013; Zheng et al., 2015, 2017); however, these design events are generally not equivalent because their joint probability density values (i.e. likelihood) usually differ (Salvadori et al., 2011; Volpi and Fiori, 2012; Li et al., 2017; Yin et al., 2017). In engineering practice, it should be necessary to determine a typical design event as representative of a specific design level. For example, in Chinese engineering practice, a unique design flood hydrograph corresponding to a given design level is usually required to determine the scale of hydraulic structures (Yin et al., 2017). The most-likely design event, which theoretically has the largest joint probability density (likelihood) among all possible design events (Salvadori et al., 2011), appears to be the best representative candidate. Besides the most-likely design event, it is also necessary to identify the confidence interval for an infinite possible number of design events to provide a finite design range for designers (Volpi and Fiori, 2012; Yin et al., 2017). The most-likely design event and confidence interval for the bivariate hydrologic design under stationary conditions have been identified (Salvadori et al., 2011; Volpi and Fiori, 2012; Li et al., 2017; Yin et al., 2017; Salvadori et al., 2018); however, very few studies have focused on hydrologic designs with higher dimensions under nonstationary conditions.

Therefore, the objective of the present study was to address the issue of multivariate hydrologic design applying to engineering practice under nonstationary conditions, which is achieved through the following steps. First, the nonstationary multivariate flood distribution was constructed using a dynamic canonical vine (C-vine) copula (Aas et al., 2009), which was able to capture the nonstationarities of both marginal distributions and the dependence structure. The design criterion for the multivariate flood event was then quantified according to AAR rather than the traditional return period, since a given multivariate flood event would correspond to a unique AAR under both stationary and nonstationary conditions (Read and Vogel, 2015; Yan et al., 2017). The multivariate hydrologic design for any given AAR was estimated from the nonstationary multivariate flood distribution.

The aforementioned methods for the multivariate hydrologic design under nonstationary conditions were applied to the Xijiang River, China. The four-dimensional (4-D) multivariate flood series, including the annual maximum daily discharge, annual maximum 3-day flood volume, annual maximum 7-day flood volume and annual maximum 15-day flood volume of the Xijiang River were chosen as the case study data because they constitute the variables used for deriving the design flood hydrograph for hydraulic structures. It has been found that the natural flood processes of this river have been significantly altered by urbanization and reservoir regulation (Xu et al., 2014), but these two factors have not yet been taken into account in multivariate hydrologic design.

The next section of the present paper describes the study area and data. Section 3 presents the methods developed in this paper. The results of the case study are provided in Sect. 4. Finally, the conclusion and remarks are provided in Sect. 5.

The multivariate flood series of the Xijiang River, South China (see Fig. 1), were selected as a case study to illustrate the multivariate hydrologic design methods under nonstationary conditions. The drainage area of the Xijiang River basin (XRB) is 353 120 km2, with a river length of 2214 km. The basin falls within a humid subtropical monsoon climate region, with the flood season extending from May to October; therefore, floods have always been a serious natural hazard within the basin.

The calculation of design floods in China involving the derivation of flood hydrographs for hydraulic structures requires not only flood peak but also flood volumes with different durations, such as 3, 7, 15 and 30 days (Ministry of Water Resources of People's Republic of China, 1996; Xiao et al., 2009; Xiong et al., 2015; Li et al., 2017). For a large catchment such as the XRB, the duration of a flood process is usually longer than 10 days. Therefore, the annual maximum daily discharge (Q1), annual maximum 3-day flood volume (V3), annual maximum 7-day flood volume (V7) and annual maximum 15-day flood volume (V15) of the Xijiang River were defined as the multivariate flood series Q1,V3,V7,V15 for deriving design flood hydrographs. The flood data were from 1951 to 2012 and observed at the Dahuangjiangkou gauge located at the main stream of the Xijiang River, draining a total catchment area of 294 669 km2, approximately 83 % of the total area of the XRB.

Rapid urbanization over recent decades has resulted in increasing river regulation projects built in the XRB, such as artificial levees for protecting urban areas from river flooding. As a result, flood flow has become increasingly constrained to the channel rather than overflowing to the floodplain, resulting in an increase in the observed river flood flow (Xu et al., 2014). For the purpose of flood control and hydropower generation, it is hard to find a river which is not impacted by reservoirs, particularly in rapidly developing China. Reservoir regulation has become an increasingly significant factor affecting flood processes of the XRB and should be seriously considered within downstream flood risk analysis and hydrologic design, particularly after 2006, when two reservoirs with considerable flood control capacities were put into operation. These are the Longtan and Baise reservoirs, with flood control capacities of 5×109 m3 and 1.64×109 m3 and catchment areas of 98 500 and 9600 km2, respectively. Climate change will likely result in flood nonstationarity by altering climatic conditions of the basin. Climatic conditions dominating flood processes in the XRB, such as extreme precipitation, appear to have been stationary over the past decades (Yang et al., 2010). Therefore, the current study introduced only urbanization and reservoir regulation as the potential driving forces of flood nonstationarity and ignored the effect of climate change.

The effect of urbanization on flood processes was quantified using the urban population (Pop). Given the unavailability of urban population data at the basin scale and the fact that the vast majority of cities in the XRB are distributed in Guangxi province, we used urban population data for Guangxi province to represent those of the XRB. The annual urban population data during the observation period were obtained from the China Compendium of Statistics 1949–2008 (Department of Comprehensive Statistics of National Bureau of Statistics, 2010) and the website of the National Bureau of Statistics of China (http://www.stats.gov.cn/tjsj/ndsj/, last access: 20 March 2019). The present study assumed the design life period for hydraulic structures to be from 2013 to 2100. The urban population over the design life period was estimated based on the predicted growth rate of China's urban population reported by He (2014). The reservoir index (RI), which depends on the catchment area and flood controlling capacities of reservoirs, was used to quantify the effects of reservoir regulation on flood processes (López and Francés, 2013). As shown in Table 1, two reservoirs with flood control functions have been completed during the observation period from 1951 to 2012, and a further two are planned for operation during the design life period. Figure 2 illustrates the evolution of the urban population and reservoir index during both the observation and design life periods.

The present study included the following methods: (1) the nonstationary multivariate flood distribution based on a dynamic C-vine copula, allowing for both time-varying marginal distributions and a time-varying dependence structure, and (2) estimation of the multivariate hydrologic design associated with AAR under nonstationary conditions. To correspond to the case study in this paper, the multivariate flood series consisting of the annual maximum daily discharge (Q1), annual maximum 3-day flood volume (V3), annual maximum 7-day flood volume (V7) and annual maximum 15-day flood volume (V15) were chosen to illustrate the multivariate design methods under nonstationary conditions. It is worth noting that the proposed methods can be extended to other multivariate flood series, such as those consisting of flood peak, flood volume and flood duration.

The statistical characteristics of both the marginal distributions and the dependence structure of multivariate flood variables can vary with time under nonstationary conditions. It is possible that the multivariate flood distribution estimated from the historical information will not reflect the statistical characteristics of flooding in the future. As a result, the stationary-based hydrologic design would not be able to deal with potential hydrologic risks of hydraulic structures. It is necessary for hydrologic designers to take into account the physical driving forces (such as human activates and climate change) responsible for the nonstationarities of multivariate flood variables.

The present study introduced possible methods for addressing multivariate hydrologic design for application in engineering practice under nonstationary conditions. A dynamic C-vine copula allowing for both time-varying marginal distributions and time-varying dependence structure was developed to capture the nonstationarities of a multivariate flood distribution. The multivariate hydrologic design under nonstationary conditions was estimated by specifying the design criterion by average annual reliability. The most-likely design event and confidence interval were identified as the outcome of the multivariate hydrologic design. Multivariate flood series Q1,V3,V7,V15 from the Xijiang River were chosen as a case study, with the main findings given below.

Urbanization and reservoir regulation were found to be the driving forces responsible for the nonstationarities of both the marginal distributions and dependence structure of the multivariate flood series Q1,V3,V7,V15. The growth of the urban population generally resulted in an increased mean value of the individual flood series, whereas it weakened the dependence of Q1,V3,V7,V15. The increasing reservoir index had the opposite effect on the individual flood series as well as their dependence. Under a given average annual reliability, the OR exceedance probability yielded the largest design values, followed by the Kendall and the AND exceedance probabilities. Nonstationarities in both marginal distributions and dependence structure affected the outcome of the multivariate hydrologic design. It is the marginal nonstationarities that played a dominant role in affecting the multivariate hydrologic design.

There are two remarks that can be made that are related to the practical implications of the hydrologic design methods developed in the current study that are detailed below.

The first remark relates to the length of observed flood data required for multivariate and nonstationary hydrologic designs. In theory, sufficiently long observed flood data (or other extreme-value data) are required to derive robust estimations of the distribution parameters and the correct hydrologic design values (Zheng et al., 2018). However, in reality, most data series are limited in length, thus forcing us to use what we have at hand to do research or design works without fulfilling the theoretical assumptions or requirements. Some recent studies suggested that univariate flood frequency analysis under stationary conditions usually requires flood data with a continuous period of at least 30 years (Ministry of Water Resources of People's Republic of China, 1996; Engeland et al., 2018; Kobierska et al., 2018). However, determining a definitive answer to the length of observed flood data that is required for flood frequency analysis under multivariate and/or nonstationary settings poses a challenge, since this issue has not yet been fully addressed. However, it is certain that multivariate and nonstationary hydrologic designs naturally require datasets of longer length, since they usually contain more parameters to be estimated.

The second remark related to the trade-off between reducing estimation bias and increasing model uncertainty. Nonstationary models generally improve performance in fitting observation data by reducing estimation bias (Jiang et al., 2015b), but this is usually achieved at the expense of increasing model complexity, such as adding more model parameters and introducing more nonstationary covariates, which might induce additional sources of model uncertainty (Serinaldi and Kilsby, 2015; Read and Vogel, 2016). A careful balance between the model fitting effect and the model complexity should be maintained in practice when employing multivariate and nonstationary hydrologic design by keeping in mind the following two points: (1) the multivariate and nonstationary models should remain effective but should also be kept as simple as possible to avoid overfitting, and (2) to ensure a robust relationship between the distribution parameters and the explanatory covariates, the chosen covariates should be physically related to the flood processes and supported by a well-defined cause–effect analysis.