## 1. Introduction

The adequate estimation of the magnitude of flood events is essential for the proper design and management of hydraulic structures, as well as for flood risk assessment. However, required flood data are often unavailable at the target site. Regional flood frequency analysis (RFFA) is then used for information transfer from gauged sites to the ungauged target site.

A number of regionalization procedures have been proposed for the estimation of flood quantiles at ungauged sites. They may be broadly classified into three main approaches: regression methods, index-flood-based methods, and geostatistical methods (see Salinas et al. (2013); Ouarda (2013), and references therein). Regression methods associate flood runoff with catchment descriptors. Index-flood-based methods assume that the flood distribution function is the same for all sites within a homogeneous region once scaled by a location parameter. Geostatistical methods consider the spatial correlation of flood runoff between catchments. These three types of approaches were considered and compared, for instance, in Ouarda et al. (2008). Recently, a number of studies have addressed the estimation of regional flood quantiles in a multivariate framework, that is, by considering not only the flood peak but also the hydrograph volume (e.g., Chebana and Ouarda 2009; Requena et al. 2016a).

All the aforementioned regionalization methods, referred to herein as “traditional” methods, have in common a prior processing and aggregation step of the regional information before estimating flood quantiles at the ungauged site. This step consists of computing the parameters, quantiles, or correlation values at the gauged sites in order to be able to estimate flood quantiles at the ungauged site through these aggregated at-site values, instead of directly using the whole available daily observed streamflow series. Although this aggregation is related to the extraction of relevant information for a given regional analysis, it implies the need for repeating the process if additional quantiles and/or additional regional analysis are required. Thus, this can be seen as a certain loss of regional information.

Over the last few years, several studies have focused on the development of improved estimates of streamflow series at ungauged sites (e.g., Parajka et al. 2013), which is essential for a number of water resources management activities. Daily streamflow series at ungauged sites may be estimated by hydrological models (rainfall–runoff) that simulate the flood generation process in the catchment, or by statistical approaches that use available streamflow series at gauged sites (e.g., Archfield et al. 2013; Ssegane et al. 2013). Hydrological models have the advantage of being useful for understanding and representing the hydrological processes in the catchment. However, they have the drawback of being demanding in terms of data and computational time (e.g., Requena et al. 2016b). They require reliable rainfall data input, timely calibration of parameters for the gauged sites, and regionalization of parameters at the ungauged site for which different methods may be considered (e.g., see Zelelew and Alfredsen 2014). Statistical approaches such as 1) the drainage-area ratio (DAR) method (Stedinger et al. 1993) and 2) the regional flow duration curve (FDC)-based methods (e.g., Mohamoud 2008; Shu and Ouarda 2012; Smakhtin et al. 1997) have been gaining popularity mainly because they are easier to apply.

The DAR method has gained widespread acceptance because of its simplicity. It assumes that the ungauged target (so-called destination) site and its donor (so-called source) site share the same daily streamflow series scaled by the ratio of their areas. A number of studies have focused on the improvement of this method (e.g., Farmer and Vogel 2013; Farmer et al. 2015). Regional FDC methods are based on the relation between flow values and the percentage of time that they are equaled or exceeded. The FDC provides a useful summary of flow variability at a given site (e.g., Vogel and Fennessey 1994). Such information is relevant for several applications, such as water supply assessment or hydropower generation. Then, the estimate of the FDC at an ungauged site, combined with a proper method for transferring the streamflow series from source sites, can be used for the estimation of daily streamflows at the ungauged site (e.g., Mohamoud 2008; Ssegane et al. 2013). A number of studies have proposed, applied, and compared regional methods for estimating the FDC at ungauged sites (e.g., Castellarin et al. 2004, 2007; Fennessey and Vogel 1990; Mendicino and Senatore 2013; Mohamoud 2008; Li et al. 2010; Ssegane et al. 2013; Zhang et al. 2014, 2015).

Studies comparing DAR-based and regional FDC-based methods reported an overall better performance of the FDC-based methods, either considering one or multiple source sites (e.g., Mohamoud 2008). For both types of methods, multiple source sites are found to lead to better results (e.g., Ergen and Kentel 2016; Hughes and Smakhtin 1996; Patil and Stieglitz 2012).

The aim of the present study is to take advantage of the large body of literature dealing with streamflow estimation at ungauged sites and the recently proposed methods for improving these estimations, in order to propose a different approach or path to conduct RFFA. The approach, named as regional streamflow-based frequency analysis (RSBFA) here, was suggested in Ouarda (2016). It consists of regionally estimating the streamflow series at the ungauged site to later estimate any quantile through local frequency analysis. Therefore, it is based on the combination of different existing procedures. The RSBFA approach is formally presented and applied to a flood case study in the present paper. Its application to a low-flow case study is shown in Requena et al. (2017, manuscript submitted to *J. Hydrology*).

The RSBFA approach applied to floods consists first of regionally estimating the daily streamflow series at the destination site and extracting its annual or seasonal maximum peak flow series. Then, a local flood frequency analysis is applied on the maximum peak flow series to obtain the desired flood quantiles. The regionally estimated daily streamflow series is obtained by using daily streamflow information at gauged sites. Thus, the available hydrological information is not a priori aggregated as in the case of traditional RFFA methods. This avoids the loss of regional hydrological information, and at the same time provides more information at the ungauged site. Unlike regression methods, it avoids the local estimation of the desired flood quantiles at the gauged sites. An overview of the philosophies and different paths taken by traditional RFFA methods and the proposed RSBFA approach is illustrated in the diagram of Fig. 1. It is important to specify that the regional estimation of streamflow series is often carried out at ungauged sites for a number of objectives. The first step of the proposed procedure is hence often already available. This makes the procedure even easier to carry out since it becomes limited to the second step, which is basically a simple at-site frequency analysis procedure. Therefore, if the proposed path is applied, efforts already spent in estimating daily streamflow series at ungauged sites may also be valued when performing regional frequency analysis.

Specifically, as a first step in the present study, the FDC-based method proposed by Shu and Ouarda (2012) is considered for the regional estimation of daily streamflow series at ungauged sites (section 2). This method is used because of its reported good performance and its simplicity, as it does not consider complicated mathematical or statistical tools. In this study, a more elaborated procedure is applied to ensure decreasing monotonicity over the point-wise FDC estimated at the ungauged site. It consists of fitting a smoothing curve to the point-wise FDC (Ouarda et al. 2010). The RSBFA approach is applied to a flood case study in the province of Quebec, Canada. Its performance is assessed through a jackknife (or leave one out) procedure and compared to the performance of traditional regional methods applied to the same case study in previous studies. Results are also broadly compared with the ones obtained in the literature for cold regions (Salinas et al. 2013).

The present paper is organized as follows. The RSBFA approach is presented in section 2. The criteria used for evaluating its performance are introduced in section 3. The case study is described in section 4. The results obtained through the application of the RSBFA approach to the case study and the comparison with the results obtained by traditional regionalization methods are shown in section 5. The discussion of the findings is given in section 6, and the conclusions are summarized in section 7.

## 2. Proposed approach

The RSBFA approach proposed for estimating flood quantiles at a destination site consists of two main steps: (i) regional estimation of daily streamflow series at the destination site and (ii) estimation of flood quantiles at the destination site by local frequency analysis. An overview of the RSBFA approach is illustrated in Fig. 2. In this study, step (i) is performed following the approach of Shu and Ouarda (2012), which can in turn be divided into two substeps: (i.a) regional estimation of FDC streamflow quantiles at the destination site and (i.b) transfer of daily streamflow series from source sites to the destination site.

Scheme of the RSBFA approach.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Scheme of the RSBFA approach.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Scheme of the RSBFA approach.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

FDC “streamflow” quantile estimation in step (i.a) is empirical and just done for a few fixed streamflow quantiles. Its estimation is regression based, in a similar way as in traditional regional regression. However, the *whole streamflow* series instead of only the maximum peak flow series is used. On the other hand, “flood” quantile estimation by a local frequency analysis in step (ii) is carried out based on the *maximum peak flow series* extracted from the regionally estimated streamflow series. This second quantile estimation allows obtaining a whole range of flood quantiles by fitting a probability distribution to the estimated maximum peak flow series. The advantage of this “iterative” quantile estimation procedure in comparison to traditional regression-based methods is that it allows obtaining any desired flood quantile without the need of previously identifying the return period it corresponds to during the regional procedure. It also avoids the need to repeat the regional procedure if a different application is sought (e.g., low flows instead of floods), as all the regional information is included in the estimated streamflows.

### a. Regional estimation of daily streamflow series at the destination site

This first step of the RSBFA approach is related to the approach of Shu and Ouarda (2012), which entails the estimation of the FDC at the destination site, and the transfer of daily streamflow series from the source sites to the destination site.

#### 1) Regional estimation of FDC streamflow quantiles at the destination site

Shu and Ouarda (2012) presented a regression-based logarithmic interpolation method for estimating the FDC at a destination site, which is the approach considered in this study. The method is summarized below. First, the FDC at each gauged site is constructed from observed daily streamflow series. Second, streamflow quantiles at a number of fixed percentile points are specifically obtained for the FDC at each gauged site. In particular, 17 unevenly spread percentile points are considered for a proper characterization of the FDC. Third, streamflow quantiles at these fixed percentile points are estimated at the destination site by applying a regional regression equation to each streamflow quantile. The aim is to build a point-wise FDC at the destination site. Fourth, the decreasing monotonicity of the estimated FDC over the fixed percentile points is assured by applying logarithmic interpolations or extrapolations when such a condition is not initially fulfilled. Fifth, intermediate points of the FDC at the destination site are estimated through logarithmic interpolation to provide a smooth FDC.

Before detailing the procedure, it is important to highlight that the aforementioned fourth and fifth steps are modified in the present study from those in Shu and Ouarda (2012). In this regard, a smoothing curve is fitted to the estimated point-wise FDC when its decreasing monotonicity is not initially fulfilled (Ouarda et al. 2010). Accordingly, the intermediate points of the FDC to which a smoothing curve is fitted are obtained through the formula of the smoothing curve instead of by logarithmic interpolation. These changes are introduced to provide a more objective procedure to ensure the decreasing monotonicity of the FDC. More details are given below.

##### (i) Construction of the FDC at each gauged site

*Q*

_{l}in descending order, with

*l*= 1, 2, …,

*n*, where

*n*is the data length that is the number of days for which streamflows are recorded at the given site. Thus,

*Q*

_{1}is the largest event and

*Q*

_{n}is the smallest one. The plotting position

*p*

_{l}that is the probability of exceedance associated with

*Q*

_{l}, is obtained with the Weibull plotting formula (e.g., Fennessey and Vogel 1990):

*x*axis and

*Q*

_{l}in the

*y*axis.

##### (ii) Estimation of the streamflow quantiles at the fixed percentile points for the FDC at each gauged site

The streamflow quantiles *Q*_{p} estimated at the 17 fixed percentile points *p* = 0.01%, 0.1%, 0.5%, 1%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 99%, and 99.99% are directly estimated from the streamflow record at the given gauged site. Following Eq. (1), at least 27 years (around *n* = 9855 days) of data are required to obtain streamflow quantiles at 0.01% and 99.99%. Then, if the available data length at a given gauged site is less than 27 years, *Q*_{0.01%} is estimated through logarithmic extrapolation, and *Q*_{99.99%} is assigned as the lowest observed streamflow value. For instance, for a 10-yr data length, *Q*_{99.99%} will have the same value as *Q*_{99.97%}.

##### (iii) Estimation of the streamflow quantiles at the fixed percentile points by applying regional regression equations to build the point-wise FDC at the destination site

*Q*

_{p}at the gauged sites. Then, a regional regression equation is applied (e.g., Mohamoud 2008), which is commonly logarithmically transformed as

*Q*

_{p}are the streamflow quantiles at the gauged sites for a given percentile point

*p*and (

*V*

_{1}, …,

*V*

_{k}) are the descriptors selected by stepwise regression for such a percentile point. The regional parameters (

*b*

_{0},

*b*

_{1}, …,

*b*

_{k}) are then obtained to be used for the estimation of

*Q*

_{p}at the destination site by replacing its corresponding catchment descriptor values in Eq. (2). For further information regarding regression methods in regional frequency analysis, the reader is referred, for instance, to Ouarda et al. (2000). It is important to note that the FDC streamflow quantiles estimated at the destination site through these regional regression equations are empirical and cannot be used for extrapolation beyond the observed series. Hence, flood quantiles at the destination site will be later obtained through a local flood frequency analysis on the maximum peak flow series extracted from the estimated streamflow series (see section 2b).

##### (iv) Assuring decreasing monotonicity for the point-wise FDC at the destination site

*a*,

*b*,

*c*, and

*d*are the parameters of the function. They are estimated by a least squares approach, where

*Q*

_{p}is constrained to be larger than zero. It can be proven that if

*a*< 0,

*b*> 0, and

*c*∈ [0, 1] the function is always decreasing.

##### (v) Estimation of intermediate points of the point-wise FDC at the destination site to provide values required during the streamflow transfer

*Q*

_{p}is the streamflow quantile to be estimated at the intermediate percentile point

*p*, and

*m*− 1 and

*m*are the existing points in the FDC which are located at its left and right, respectively.

#### 2) Transfer of daily streamflow series from source sites to the destination site

*p*of the streamflow value at the source site is obtained from its FDC. Then,

*p*is searched into the FDC at the destination site to obtain its corresponding streamflow value [see Fig. 2, diagram (i)]. Ssegane et al. (2013) supported naming this approach “the streamflow separation technique” to highlight the estimate of streamflow series as a combination of streamflow magnitude from the estimated FDC and temporal sequence from the daily streamflow series at the source site. When

*N*source sites are considered, the streamflow value estimated from each one,

*Q*

^{d,j}, is weighted by

*w*

_{j}for obtaining the streamflow at the destination site

*Q*

^{d}as

*d*in order to obtain the daily streamflow series at the destination for the time period being considered.

Several studies focused on the selection of the source site(s) for transferring streamflow series to the destination site, as their identification is essential for achieving a suitable prediction at the destination site. A common finding is that more than one site is always preferable and that a large number of sites is counterproductive. In this regard, three (e.g., Ergen and Kentel 2016), four (e.g., Shu and Ouarda 2012), or five (e.g., Hughes and Smakhtin 1996; Patil and Stieglitz 2012) source sites are usually considered. Such a selection is based on spatial proximity, similarity of catchment descriptors, and/or correlation of streamflows (e.g., Archfield and Vogel 2010; Archfield et al. 2013; Shu and Ouarda 2012). In the present study, four source sites are selected based on spatial proximity between destination and source sites, following Shu and Ouarda (2012). Spatial proximity is supported by Hughes and Smakhtin (1996), who suggested that source sites could be those in the same river or tributaries, while a high distance may entail a negative impact on predictability (e.g., Ssegane et al. 2013). The elaborated correlation-based approach for selecting source sites, called the map correlation method (Archfield and Vogel 2010; Ergen and Kentel 2016), is not considered in this study. This is because a dense gauging network entailing a long common record period is needed for its application, while the present study aims to provide a general method that is also easy to apply in practice. The focus of the present study is not the development of improved daily streamflow methods, but rather the exploration of a different avenue for regional flood quantile estimation at ungauged sites.

*w*

_{j}assigned to the four selected source sites are then estimated by a weighting scheme also based on spatial proximity in the present study:

*d*

_{j}is the similarity distance measure between the destination site and the source site

*j*, and LAT and LONG are the latitude and longitude, respectively. It should be mentioned that other weighing schemes could be applied by considering different catchment descriptors. For instance, Patil and Stieglitz (2012) suggested considering spatial proximity, climatology, geology, and topography for transferring streamflows. However, a preliminary analysis performed for the present case study showed that a weighting scheme based on geographical distance provides the best results, which supports the finding of Shu and Ouarda (2012) for different sites located in the same area of study.

### b. Estimation of flood quantiles at the destination site by a local frequency analysis

A local flood frequency analysis is performed at the destination site by using the daily streamflow series regionally estimated through the procedure described in section 2a. For this purpose, the seasonal or annual maximum peak flow series is extracted from this daily streamflow series. The aim is to allow for a proper estimation of moderate and/or high flood quantiles at the destination site. Note that moderate flood quantiles may be used, for example, in urban drainage studies, or for the design of temporary structures. Also note that other approaches exist when specifically focusing on high quantiles (e.g., Naveau et al. 2014). Distributions commonly used in flood frequency analysis are then considered for fitting the maximum peak flow series. The candidate distributions are the generalized extreme value (GEV), Gumbel (G), two-parameter lognormal (LN2), three-parameter lognormal (LN3), gamma (GA), Pearson type 3 (P3), and log-Pearson type 3 (LP3). Several parameter estimate methods, such as the method of moments, L-moments, or maximum likelihood are considered. The Bayesian information criterion (BIC) proposed by Schwarz (1978) is used as an objective criterion to select the best model for fitting the data while integrating parsimony considerations (e.g., Laio et al. 2009). The selected distribution fitted to the maximum peak flow series is then used for estimating the desired flood quantiles corresponding to a given return period. Goodness-of-fit tests and graphical representations can also be used in the identification of the appropriate distribution (for additional information regarding distribution selection, see, e.g., Laio 2004; Requena et al. 2016a).

## 3. Evaluation of the proposed approach

*q*

_{i}is the local specific flood quantile, and

*N*

_{t}is the total number of sites in the study region. The local specific flood quantile is estimated at site by using the observed data at the given site. A Q–Q plot of local versus regionally estimated specific flood quantiles is also presented for a visual analysis. Finally, the predictive performance of the RSBFA approach is compared with that of traditional RFFA approaches applied on the same case study. Its performance is also broadly compared with the results reported in the literature for similar regions.

## 4. Case study

The case study considered for the application of the RSBFA approach is the hydrometric station network of the southern part of the province of Quebec, Canada. Hydrological data from a total of 151 sites managed by the Ministry of the Environment of Quebec (MENVIQ) services are used. These sites were selected based on the following criteria: present a natural flow regime; have a historical flood record of at least 15 years; and pass homogeneity, stationarity, and independence tests. The 151 sites are located between 45° and 55°N (Fig. 3), and their catchment area ranges from 200 to 100 000 km^{2}. This case study has already been used for the application and testing of a number of regional frequency analysis procedures in the literature (see, e.g., Ouarda and Shu 2009). Its adoption allows for a direct comparison with a number of commonly adopted RFFA methods.

Location of the 151 hydrometric sites in southern Quebec, Canada. Sites are also marked according to the number of years of daily streamflow values regionally estimated for the 30-yr study period.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Location of the 151 hydrometric sites in southern Quebec, Canada. Sites are also marked according to the number of years of daily streamflow values regionally estimated for the 30-yr study period.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Location of the 151 hydrometric sites in southern Quebec, Canada. Sites are also marked according to the number of years of daily streamflow values regionally estimated for the 30-yr study period.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

A local flood frequency analysis was performed at each of the selected sites by Kouider (2003). Flood quantiles corresponding to several return periods were obtained by fitting the best distribution to the observed records, based on model selection criteria. In the present study, we focus on spring floods that are mainly generated by snowmelt. In general, snowmelt runoff occurs from 1 April to 30 June in the study region of the province of Quebec (Ben Aissia et al. 2012). We consider specific quantiles, which are quantiles divided by the catchment area, in order to avoid scale effects (e.g., Chokmani and Ouarda 2004). Thus, two typical and representative spring flood quantiles are considered: the specific quantiles corresponding to return periods of 10 and 100 years (*q*^{10} and *q*^{100}). They are also selected for comparison purposes with previous studies. Corresponding local quantiles are required for the assessment of the performance of the RSBFA approach through Eqs. (8) and (9).

Physiographical and meteorological data were also available for the selected sites. The physiographical variables were extracted from both the MENVIQ hydrological database and the topographic digital maps of Quebec. The physiographical variables considered are catchment area (BV), catchment mean slope (PMBV), fraction of the catchment controlled by lakes (PLAC), fraction of the catchment occupied by forest (PFOR), main channel length (LCP), main channel slope (PCP), altitude (ALT), LAT, and LONG. The meteorological variables were estimated by interpolating historical data from the MENVIQ meteorological network across the province of Quebec. The meteorological variables considered are annual mean total precipitation (PTMA), annual mean degree days below 0°C (DDBZ), annual mean liquid precipitation (PLMA), annual mean solid precipitation (PSMA), and mean snow level on 30 March (MNS30M). A summary of the descriptive statistics of the hydrological, physiographical, and meteorological variables is presented in Table 1.

Summary of the descriptive statistics of the hydrological, physiographical and meteorological variables. Asterisks indicate the five descriptors considered in previous regional flood frequency analysis for the case study.

This database has been specifically used in a number of flood regional studies by considering five of these catchment descriptors (e.g., Chokmani and Ouarda 2004; Shu and Ouarda 2008; Wazneh et al. 2013). The five descriptors consist of the three physiographical variables BV, PMBV, and PLAC and the two meteorological variables PTMA and DDBZ. In this regard, the RSBFA approach is applied in two situations. First, it is applied by only using these five descriptors, with the aim of specifically comparing its results with those of traditional regional analysis for the same case study. Then, it is also applied by considering all the available predictors in order to assess the robustness of the approach in terms of the effect of the variables.

## 5. Results

### a. Regional estimation of daily streamflow series at the destination site

First, the FDC at the destination site is estimated by following the procedure explained in section 2a(1). The catchment descriptors selected by stepwise regression for characterizing each of the 17 streamflow quantiles are shown in Table 2. Note that in this first application of the procedure only the five catchment descriptors BV, PMBV, PLAC, PTMA, and DDBZ are used for comparison purposes, yet a larger number could be considered in practice. For each destination site, the streamflow quantile corresponding to a given percentile point is obtained by the regression expression in Eq. (2). As it can also be seen in Table 2, the coefficient of determination *R*^{2} estimated by applying a jackknife procedure is overall very high for all streamflow quantiles. Slightly lower values are obtained for *Q*_{0.01%} and *Q*_{99.99%} because of the higher uncertainty associated with extreme quantiles.

Catchment descriptors selected by stepwise regression for characterizing the streamflow quantiles *Q*_{p} and associated coefficient of determination *R*^{2} estimated by a jackknife procedure.

The decreasing monotonicity of the point-wise FDC estimated at the destination site is ensured by fitting the smoothing curve in Eq. (3) when such a condition is not preserved. For the present case study, it is necessary to fit the smoothing curve at 20 of the 151 sites. An example of the observed, estimated, and smoothed FDC at a given destination site is shown in Fig. 4. The summary of the descriptive statistics of the smoothing curve parameters is shown in Table 3.

Example of the observed, estimated, and smoothed FDC at a given destination site: whole FDC, zoom to the upper extreme (high flows), and zoom to the lower extreme (low flows).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Example of the observed, estimated, and smoothed FDC at a given destination site: whole FDC, zoom to the upper extreme (high flows), and zoom to the lower extreme (low flows).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Example of the observed, estimated, and smoothed FDC at a given destination site: whole FDC, zoom to the upper extreme (high flows), and zoom to the lower extreme (low flows).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Summary of the descriptive statistics of the smoothing curve parameters.

Then, the nonlinear spatial interpolation method is used for transferring streamflows from the source sites to the destination site following the procedure explained in section 2a(2). Intermediate points of the FDC at a given site are obtained by Eq. (4) or by Eq. (3) if the smoothing curve was previously fitted. A 30-yr study period is considered for estimating daily streamflow values at the destination site, in order to have enough data to be able to later perform a local flood frequency analysis. The 30-yr study period ranges from 1 January 1971 to 31 December 2000. The record of available daily streamflow series for each of the 151 sites is shown in Fig. 5a. It can be seen that although the 151 sites are gauged sites, they have missing data over the selected study period (empty spaces in Fig. 5a). Indeed, 13 of the 151 sites are ungauged for such a period (see also Fig. 3). Note that daily streamflow series at these sites will be regionally estimated by the proposed procedure. Hence, the quality of the flood quantiles obtained by the RSBFA approach will be later assessed for the 151 sites of the case study (section 5c).

Days of the 30-yr study period with (a) observed streamflow values and (b) regionally estimated streamflow values for each site. A given day is expressed as year-month-day, for example, 19710101 corresponds to 1 Jan 1971.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Days of the 30-yr study period with (a) observed streamflow values and (b) regionally estimated streamflow values for each site. A given day is expressed as year-month-day, for example, 19710101 corresponds to 1 Jan 1971.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Days of the 30-yr study period with (a) observed streamflow values and (b) regionally estimated streamflow values for each site. A given day is expressed as year-month-day, for example, 19710101 corresponds to 1 Jan 1971.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

A given daily streamflow value is obtained at the destination site by combining the daily streamflow estimates given by its four source sites [Eq. (5)]. The values estimated by these four source sites, which are selected by spatial proximity, are weighted as in Eq. (6). It is important to mention that, if for a given day there is a missing value at either one or several source sites, such weights are re-estimated by only considering the source sites with an available daily streamflow value. The set of days over the study period for which a regionally estimated streamflow value is obtained by applying this procedure is shown in Fig. 5b. As can be seen, the regional procedure allows estimating practically the whole daily streamflow series for the 30-yr study period at the 151 sites in the study region. Specifically, more than 26 years of daily streamflow values are estimated for around 90% of the sites. The aforementioned 13 ungauged sites for the study period are included in this 90%. Between 18 and 26 years of daily streamflow values are estimated for slightly more than 5% of the sites. Just a limited period between 9 and 14 years is estimated at slightly less than 5% of the sites. This is due to the limited observed daily streamflow series of their corresponding source sites. This site classification according to the number of years of regionally estimated daily streamflows is illustrated in Fig. 3.

In the present study, the regional estimation of daily streamflow series corresponds to intermediate results. The performance of these estimated series regarding the observed values is in the same order of magnitude as the one in Shu and Ouarda (2012), for which different sites and study period were considered (not shown). Note that the focus of Shu and Ouarda (2012) was to estimate streamflow series, but not flood quantiles.

### b. Estimation of flood quantiles at the destination site by a local frequency analysis

The spring maximum peak flow series is extracted from the regionally estimated daily streamflow series at each of the 151 sites. The distribution and parameter estimate method selected as the best model for fitting the series following section 2b are shown in Fig. 6. Note that only model selection is considered with the aim of using the same criterion as when local quantiles were obtained. As a result, the GA distribution is selected for 92 sites (61% of the sites). The LN2 distribution is chosen for 29 sites (19%). The G distribution is selected for 15 sites (10%), and the GEV distribution is selected for 8 sites (5%). The LN3 and LP3 distributions are selected for one and six sites, respectively, which is a total of around 5% of the sites. The selected distribution at each site is then used for estimating the specific spring flood quantiles *q*^{10} and *q*^{100}. A summary of the statistics of the estimated and local specific spring flood quantiles is shown in Table 4. Related Q–Q plots are shown in Fig. 7a. From these results it is observed that estimated and local specific quantiles are overall of the same order of magnitude, with underestimation of high specific quantile values. Other regionalization studies on the same case study obtained similar results. This may be attributed to the existence of small catchments for which catchment size is in turn underestimated (Chokmani and Ouarda 2004).

Distributions selected as the best model for fitting the spring maximum peak flow series regionally estimated at the 151 sites of the study region.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Distributions selected as the best model for fitting the spring maximum peak flow series regionally estimated at the 151 sites of the study region.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Distributions selected as the best model for fitting the spring maximum peak flow series regionally estimated at the 151 sites of the study region.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Summary of the statistics of the specific and absolute, local and estimated, spring flood quantiles.

Q–Q plot for 10- and 100-yr return period spring flood quantile: (a) specific quantile *q* and (b) absolute quantile *q*_{a}.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Q–Q plot for 10- and 100-yr return period spring flood quantile: (a) specific quantile *q* and (b) absolute quantile *q*_{a}.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Q–Q plot for 10- and 100-yr return period spring flood quantile: (a) specific quantile *q* and (b) absolute quantile *q*_{a}.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

### c. Evaluation of the proposed approach

The results of the RSBFA approach regarding RB [Eq. (8)] and RRMSE [Eq. (9)] are shown in Table 5 as approach XVIII. These values are compared with results obtained by traditional regionalization methods (Table 5). Approaches from I to XVII are considered in this comparison due to entailing different regionalization methods that are based on the same 151 sites, the same five catchment descriptors (see Table 1), and the use of spring flood data. These approaches may be broadly classified as regression-based methodologies. They considered specific quantiles, with the exception of approaches VIII–X, where absolute quantiles were used. The comparison of both kinds of results is allowed as a result of using relative error measures such as RB and RRMSE. Overall, the RB^{T} and RRMSE^{T} values of the RSBFA approach for both return periods *T* = 10 and 100 years are comparable with those of traditional regional analysis. The RSBFA approach leads to the best results for RB, with the minimum absolute values. RRMSE^{100} is equal to or less than the corresponding value of 10 of the 17 approaches considered. RRMSE^{10} is equal to or less than the corresponding values of 7 of the approaches. Its graphical comparison is presented in Fig. 8 by plotting the RRMSE of the RSBFA approach on the box plot of the corresponding RRMSE values of the traditional approaches. As can be seen, the results of the RSBFA approach are within the 25%–75% range of the RRMSE values of the traditional approaches and closer to the median than to the upper boundary. In particular, RRMSE^{100} presents a better performance since it is located below the median.

Comparison among the performances of the RSBFA approach and traditional regional analysis approaches. Superscript indicates values associated with a 10- or a 100-yr return period. Values are given in percentage. Values in bold indicate studies obtaining better results than the RSBFA approach numbered as XVIII.

Box plots of the RRMSE results of traditional regional analysis performed on the case study by using the five descriptors marked in Table 1 (approaches I–XVII in Table 5). Results are presented for *T* = 10- and 100-yr return periods. The points represent the RRMSE results of the RSBFA approach (XVIII and XX in Table 5). The solid rectangle represents the 25%–75% range of the RRMSE results for cold regions shown in Salinas et al. (2013) (XIX in Table 5).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Box plots of the RRMSE results of traditional regional analysis performed on the case study by using the five descriptors marked in Table 1 (approaches I–XVII in Table 5). Results are presented for *T* = 10- and 100-yr return periods. The points represent the RRMSE results of the RSBFA approach (XVIII and XX in Table 5). The solid rectangle represents the 25%–75% range of the RRMSE results for cold regions shown in Salinas et al. (2013) (XIX in Table 5).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Box plots of the RRMSE results of traditional regional analysis performed on the case study by using the five descriptors marked in Table 1 (approaches I–XVII in Table 5). Results are presented for *T* = 10- and 100-yr return periods. The points represent the RRMSE results of the RSBFA approach (XVIII and XX in Table 5). The solid rectangle represents the 25%–75% range of the RRMSE results for cold regions shown in Salinas et al. (2013) (XIX in Table 5).

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

The results of the RSBFA approach by considering the catchment descriptors in Table 1 are included as approach XX in Table 5. As expected, the inclusion of a larger set of descriptors improves quantile results, as more information is available and is being used. It results in an overall improvement in the estimation of the FDC and in the estimation of the daily streamflow series (not shown). The fact that results are better than by considering only the five aforementioned descriptors, but of the same order of magnitude, supports the robustness of the RSBFA approach in terms of the effect of the variables. Its performance is not highly affected by the amount of available information.

Based on these results, a general comparison with different traditional regional frequency analysis methods over the literature, including regression methods, index-flood-based methods, and geostatistical methods, is also provided in Fig. 8. This is based on the comparison analysis performed by Salinas et al. (2013). Specifically, the 25%–75% range of RRMSE^{100} in such a study was identified as around 45%–64% for cold regions, where the province of Quebec was included. Therefore, RRMSE^{100} = 49% obtained by the RSBFA approach by considering the catchment descriptors in Table 1 is within this range, and closer to the lower boundary. Note that RRMSE^{100} = 57% obtained when only considering the five descriptors is also within this range.

For a more detailed analysis, the at-site relative errors (*q*_{i} − *q*_{i} from Eqs. (8) and (9) are shown in Figs. 9 and 10. For comparison purposes these values are the ones obtained for *q*^{100} by using the five descriptors. A bar plot of the relative errors is shown in Fig. 9. Of the sites, 25% are found to have values equal or less than |0.10|, 55% have values equal or less than |0.25|, and 84% have values equal or less than |0.50|. The number of years of regionally estimated daily streamflows is not found to be associated with the size of the relative error, as sites with less than 26 years of estimated daily data present small relative errors (Fig. 9). Except for sites 10 and 62, all the sites that were ungauged for the 30-yr study period obtained relative errors that are less than |0.50|. It is also found that site numbers 46, 64, 66, and 148 present a very large relative error. However, these particular sites were already identified as problematic sites in previous studies (e.g., Chebana and Ouarda 2008; Chokmani and Ouarda 2004). For instance, some of them were found to have underestimated drainage areas. These relative errors are also displayed in the map of Fig. 10. Neither a location nor a catchment size pattern is found to be related to the size of the errors for the categories of catchment size and relative error considered.

Bar plot of the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Bar plot of the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Bar plot of the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Map with the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Map with the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Map with the relative errors obtained at each site for *q*^{100} by applying the RSBFA approach by considering the five descriptors marked in Table 1.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

## 6. Discussion

The RSBFA approach leads to results comparable to the ones obtained by traditional regional methods for which also more complicated mathematical and statistical tools are overall considered. The case study application illustrates the easiness by which the proposed method can be applied. Although it does not always achieve the best results (Table 5, Fig. 8), it possesses an important advantage over traditional methods. It allows for a continuous flood frequency analysis, instead of the discontinuous and limited quantile estimates provided by traditional regional approaches. Thus, if additional quantiles to those originally estimated are required, they may be easily obtained by the RSBFA approach. This is because all the regional information is contained within the estimated daily streamflow series. Then, any quantile, specific or absolute and annual or seasonal, can be directly obtained through a simple local flood frequency analysis without the need to redo the whole regional procedure. This is not the case for traditional RFFA approaches. The RSBFA approach also avoids the assumption of a common regional distribution adopted by index-flood-based methods.

For instance, if absolute quantiles are needed for designing a hydraulic structure at a given ungauged site, they can be directly obtained from the fitted distribution to the maximum peak flow series extracted from the regionally estimated daily streamflow series. Indeed, specific quantiles are simply obtained by dividing these absolute quantiles by the catchment area. If subsequently a quantile related to another return period is needed, for instance, to perform a risk assessment analysis, it can be directly obtained by using the same fitted distribution. As an illustration of the flexibility of the approach, a summary of the statistics of the absolute quantiles *q*_{a} for the case study are shown in Table 4, and the associated Q–Q plots are displayed in Fig. 7b. Also, it should be noted that RB and RRMSE values of the RSBFA approach are the same whether specific or absolute quantiles are considered. However, in other regionalization approaches such as regression methods, statistical procedures are applied on the given quantile of study, and then absolute quantiles are obtained independently of specific quantiles.

Moreover, the flexibility provided by the RSBFA approach allows also performing multivariate analysis at the ungauged site. This is because the whole information of each flood event is available. Consequently, flood characteristics such as maximum peak flow and associated hydrograph volume and duration can be directly obtained from the estimated series. An example of the hydrographs obtained by the regionally estimated daily streamflow series for a given site over a given period is shown in Fig. 11. In summary, the fact that the “ungauged” site is transformed into a “gauged” site allows performing any kind of local flood frequency analysis. This makes the RSBFA approach a very flexible approach, which opens a number of possibilities. For instance, it also facilitates the combination of local and regional information, which is often carried out using complex procedures such as Bayesian approaches (e.g., Seidou et al. 2006). It is also important to mention that, depending on data availability for a given case study, the RSBFA approach may be applied for steps other than the daily streamflow one.

Example of the hydrographs obtained for a given site and a given period through the regionally estimated daily streamflow series. Observed and regionally estimated streamflow series are shown for comparison purposes.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Example of the hydrographs obtained for a given site and a given period through the regionally estimated daily streamflow series. Observed and regionally estimated streamflow series are shown for comparison purposes.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

Example of the hydrographs obtained for a given site and a given period through the regionally estimated daily streamflow series. Observed and regionally estimated streamflow series are shown for comparison purposes.

Citation: Journal of Hydrometeorology 18, 9; 10.1175/JHM-D-16-0143.1

The performance of the proposed procedure depends on the estimate of the FDC, on the selection of the source sites, and on the selection of the probability distribution. Delineation of homogeneous regions (e.g., Castellarin et al. 2004; Mendicino and Senatore 2013; Waseem et al. 2015) could be considered for improving the estimate of the regression-based point-wise FDC at ungauged sites in section 2a(1). Additional research could also be performed for simplifying the steps related to preserve the decreasing monotonicity condition. Further research could also be dedicated to the refinement of the procedure for regionally estimating daily streamflow series at ungauged sites. In this line, the final objective would be to obtain results that are even better than those obtained by using sophisticated regional analysis procedures (see Table 5). Regarding the selection of the source sites, a more elaborate procedure, such as the map correlation method (Archfield and Vogel 2010), could be used when a dense gauging network with a long common daily streamflow record is available in the study region. Other approaches for the identification of source sites, such as nonlinear canonical correlation analysis or depth functions (see Wazneh et al. 2015), can also be considered. Finally, the selection of the probability distribution for fitting the maximum peak flow series could be based on a more detailed selection process, including model selection criteria and goodness-of-fit tests (e.g., Requena et al. 2016a). Note that only one model selection criterion (BIC) was considered in the present study for comparison purposes.

The RSBFA approach was applied to a large flood case study in the province of Quebec, Canada, which may be classified as a cold region. Its application to other types of regions (such as mountainous catchments) and/or climates would be important in order to check the generality of the conclusions. Assessing its application to dry regions would be especially relevant, since daily streamflows and flood predictability are lower for these regions (e.g., Parajka et al. 2013; Patil and Stieglitz 2012; Salinas et al. 2013). Moreover, drought and low-flow frequency analysis could be performed through the RSBFA approach. In this regard, it is important to note that the application of the RSBFA approach for the frequency analysis of other streamflow variables is possible because of the adoption of a general approach in which a full probability distribution is fitted at the local frequency analysis step. Also note that, although some low-flow quantiles may be directly obtained from the FDC, others such as the ones related to the 7- or 30-day minimum flow series can only be estimated through a low-flow frequency analysis on the estimated streamflow series. Application of the proposed methodology to a low-flow case study in the province of Quebec is presented in Requena et al. (2017, manuscript submitted to *J. Hydrology*).

## 7. Conclusions

In the present study, a different approach for conducting regional flood frequency analysis (RFFA) is presented. It consists of regionally estimating the daily streamflow at the ungauged site prior to conducting a local flood frequency analysis. The proposed approach, referred to as regional streamflow-based frequency analysis (RSBFA), has the advantage of using the whole amount of hydrological information at the gauged sites, without the prior aggregation and processing commonly carried out by traditional RFFA procedures. The RSBFA approach avoids also performing a complete at-site flood frequency analysis at each gauged site. The fact that all the regional information is included in the estimated daily streamflow series provides a very flexible and continuous approach. Hence, absolute or specific, univariate or multivariate, and annual or seasonal quantiles corresponding to any return period can be obtained through the regionally estimated daily streamflow series.

The RSBFA approach is applied to a case study in the province of Quebec, Canada, and the results are compared with those obtained by traditional RFFA methods performed on the same case study. Results are also compared with traditional regional analysis results available in the literature for cold regions. The performance of the RSBFA approach is comparable to the performances of traditional RFFA approaches, with less effort and the advantage of providing the whole daily streamflow series.

## Acknowledgments

The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Merit scholarship program for foreign students–Postdoctoral research fellowship of the Ministère de l’Éducation et de l’Enseignement Supérieur du Québec managed by the Fonds de recherche du Québec*–*Nature et technologies is gratefully acknowledged. The authors are grateful to the Editor, Dr. Andrew Wood, and to three anonymous reviewers for their comments which greatly helped improve the quality of the manuscript. The authors also acknowledge the assistance from Christian Charron and Martin Durocher.

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