Measuring the impact of climate change on flood frequency is a complex and controversial task. Identifying hydrological changes is difficult given the factors, other than climate variability, which lead to significant variations in runoff series. The catchment filtering role is often overlooked and thus may hinder the correct identification of climate variability signatures on hydrological processes. Does climate variability necessarily imply hydrological variability? This research aims to analytically derive the flood frequency distribution based on realistic hypotheses about the rainfall process and the rainfall–runoff transformation. The annual maximum peak flow probability distribution is analytically derived to quantify the filtering effect of the rainfall–runoff process on climate change. A sensitivity analysis is performed according to typical semi-arid Mediterranean climatic and hydrological conditions, assuming a simple but common scheme for the rainfall–runoff transformation in small-size ungauged catchments, i.e. the CN-SCS model. Variability in annual maximum peak flows and its statistical significance are analysed when changes in the climatic input are introduced. Results show that depending on changes in the annual number of rainfall events, the catchment filtering role is particularly significant, especially when the event rainfall volume distribution is not strongly skewed. Results largely depend on the return period: for large return periods, peak flow variability is significantly affected by the climatic input, while for lower return periods, infiltration processes smooth out the impact of climate change.

Many of the concerns about climate change are related to its effects on the hydrological cycle (Kundzewicz et al., 2007, 2008; Koutsoyiannis et al., 2009; Bloeschl and Montanari, 2010), and more specifically, its impact on freshwater availability and flood frequency (Milly et al., 2002; Kay et al., 2006; Allamano et al., 2009). However, results from recent studies about climate change impacts on flood frequency have not been conclusive (Kay et al., 2006). Indeed, detecting changes in flood frequency is not easy, because there are factors other than climate variability that may lead to significant changes – for instance, spatial variability of watershed properties or changes in the channel network geometry and land-use change (Milly et al., 2002). In particular, river bed geometry alterations, even if localized, can significantly affect flood magnitude. Therefore, to better identify climate impacts, one should focus on catchments that are in close to pristine conditions (Di Baldassarre et al., 2010).

This research addresses an issue that is often overlooked and which may hinder the proper identification of climate variability effects on hydrological processes – namely, the filtering role played by catchment. In fact, runoff can be interpreted as a smoothed convolution of past and current rainfall, where smoothing is operated over the catchment contributing area and along the concentration time. Depending on the catchment's physical characteristics and meteorological conditions, smoothing may average out changes in rainfall distribution in space and time and hence cancel out climate variability. This is a key reason why climate variability effects might not be clearly visible in the hydrology response. In other words, climate variability does not necessarily imply hydrological variability. This issue has been also investigated for an urban hydrology context. For example, Andrés-Doménech et al. (2012) analysed storm tank resilience to changes in rainfall statistics, proving that the effect of climate variability on storm tank efficiency is likely to be smoothed out by the filtering effect caused by the urban catchment.

In the present study, modelling efforts are basically centred on the role of climatic variability and its effects on catchment hydrological response, with rainfall statistical properties and their future trends representing the major factors controlling flood frequency distribution. It should be noted that other factors, such as land use change, might have a more significant impact than climate change itself under certain hydrological conditions. The present research focuses on climatic impacts alone: interactions at the catchment scale between landscape characteristics (soils, vegetation and geology, for instance) and climatic properties (Troch et al., 2013), or possible climate-vegetation-soil feedbacks are not considered as they may hinder the assessment of climatic effects.

The modelling framework and simulations performed in this study focus on rainfall patterns' variability, using a suitable modelling framework to investigate the extent to which such rainfall variations can actually be buffered by a given standard hydrological catchment, with typical response parameters of a small catchment in a semi-arid Mediterranean region. Thus, heterogeneity in catchment physical properties, which has provided contrasting and sometimes contradictory results (Sangati et al., 2009), is not considered in the presented approach. Runoff statistics sensitivity to spatial heterogeneity is in principle less significant as the catchment area is smaller and therefore more homogeneous. In our case, we assume that the concentration time is short, therefore implying that the catchment area is small. Thus, the lumped modelling assumption can be considered reasonable for the purpose of the study.

To assess climatic impacts, the frequency of occurrence of peak flows is estimated by means of a derived distribution approach, which is particularly useful to obtain probability distributions of peak flows in ungauged or poorly observed basins. In such cases design floods are calculated from a hydrological model, which is driven by historical or synthetic rainfall data (Haberlandt and Radtke, 2014). The derived flood frequency analysis was also used by Gaume (2006) to investigate the asymptotic behaviour of flood peak distributions from rainfall statistical properties, highlighting the strong dependence of peak flow distribution on rainfall statistical properties, and considering a limited and reasonable hypothesis on the rainfall–runoff transformation.

Accordingly, a stochastic process is used here to model rainfall and a simple deterministic lumped model is proposed to simulate the rainfall–runoff transformation. Such an analytical approach, which has a long history of application in hydrology (see, for instance, Eagleson, 1972 and Papa and Adams, 1997), presents several advantages. The most relevant is the opportunity to analytically assess the cause–effect relationships that take place in the rainfall–runoff transformation.

However, the analytical approach requires the use of models that lend themselves to analytical developments, which are obtained by using simplified representations. Therefore our analysis, being based on the use of an analytical model, cannot account for the overall complexity of catchment processes. Consequently, a simplified representation of hydrological processes is considered herein, without including detailed effects.

Under such assumptions, the aim of this research is to quantify the actual extent to which the rainfall–runoff process actually filters the impact of rainfall variability on runoff annual maximum peak flow series. The flood frequency distribution is analytically derived for a hypothetical catchment based on plausible assumptions about the rainfall process and the rainfall–runoff transformation. Having derived the peak flow probability distribution, one may quantify the smoothing brought on by the rainfall–runoff process. A hypothetical case study is developed according to climatic and hydrological conditions typical of the Valencia region (Spain), described in Sect. 2.2. As also described later, the rainfall–runoff model proposed assumes a simple but common scheme for small, fast-responding, ungauged catchments, subjected to erratic hydrological regimes (Ferrer Polo, 1993; Soulis and Valiantzas, 2012).

We set up an analytical model to describe the river flow regime for a hypothetical catchment, based on analytical descriptions of rainfall and rainfall–runoff transformation. Under suitable assumptions which are described below, this model allows us to derive the annual maximum flood frequency distribution, depending on climate and catchment behaviour.

The analysis presented herein is an event-based approach, where each rainfall–runoff event is treated as an independent event. In the Valencia region, as in other many semi-arid locations around the Mediterranean, ephemeral rivers are closely related to small and fast-responding catchments. Such regimes, also named as “erratic regimes” according to the classification provided by Botter et al. (2013), occur when rainfall inter-arrival times are somewhat longer than the typical duration of the resulting flow pulses, as the case presented in this study. As pointed out by Andrés-Doménech et al. (2010), antecedent dry periods for the considered climate can be assumed to be exponentially distributed with a 22 h low bound and an 8-day expected mean value. With such a sporadic rainfall regime, antecedent moisture conditions are mainly related to the event itself and rainfall intensities during the initial stages of the storm, so that the assumption of independence for subsequent events is plausible. Moreover, for this type of hydrological event, direct runoff is the dominant component of the hydrograph.

To carry out this analysis, we assume that the rainfall forcing in the present climate can be modelled by a stationary model. Thus, non-stationarity can be accounted for by changing the parameters of the rainfall model at a given time when climate variability is supposed to occur. Such a change in the rainfall model parameters implies a corresponding deterministic change of rainfall statistics and therefore non-stationarity (Koutsoyiannis and Montanari, 2014; Montanari and Koutsoyiannis, 2014). Non-stationarity in the river flow is assumed to occur for the presence of the above non-stationarity in rainfall and thus is quantified through the proposed approach.

A rainfall analytical model is used to describe the occurrence of the
rainfall process over time. We adopt a stochastic rectangular-pulse model
that simulates rainfall dynamics by assuming that rainfall events occur as
independent rectangular pulses over time. Events are assumed to occur
according to a Poisson process (Madsen and Rosbjerg, 1997; Madsen et al.,
1997) and thus the probability of experiencing

The distribution function of the generalized Pareto distribution is given by

For the region that is considered in the study, convective storms usually occur during autumn, particularly in September and October, while frontal events mostly occur during winter and spring. Thus, maximum rainfall peaks occur systematically during autumn. The rainfall model that we use can potentially reproduce both frontal and convective events (see, for instance, Andrés-Doménech et al., 2010). Consequently, seasonality is not specifically accounted for. We assume that climatic variability may occur through an intensification of rainfall events, and we investigate the conditions under which it may imply or not an amplification of annual maximum floods – that is, to what extent the rainfall–runoff transformation may filter out or amplify the effects of climate variability.

To conceptualize rainfall–runoff transformation, the SCS-CN event-based
model was adopted. This model has been widely used in Spain (Ferrer Polo,
1993) and other Mediterranean countries (Soulis and Valiantzas, 2012). In
this model, runoff volume,

The original SCS model recommends a standard value

The rainfall and rainfall–runoff analytical descriptions allow for the
analytical derivation of the probability distribution function (PDF) of all
events peak flow. Assuming that no runoff occurs if

Asymptotic properties of the maximum likelihood estimators (MLEs) of the
generalized Pareto distribution (Eq. 2) such as consistency, normality and
efficiency were obtained by Smith (1984). The MLEs (

Based on the previously established assumptions, the analysis shows that the
following parameters affect the magnitude of the annual maximum peak river
flow

expected number of rainfall events per year,

shape and scale parameters,

storage capacity of the catchment,

initial abstraction of the catchment,

concentration time of the catchment

SCS peak factor

return period,

The dependence of

The sensitivity to the other climatic and catchment parameters is to be
analysed through Eq. (7). Specifically, an increase in the flood
quantile is induced by an increase in parameters

Rainfall model parameters are estimated by maximum likelihood for the
1990–2006 data series in Valencia. Resulting values are

Parameters defining the catchment are adopted in a dimensionless form. This
analysis focuses on how the production parameters influence the peak flow
statistics. Thus, the storage capacity is considered through the ratio

Peak flows are expressed per unit area (mm h

The first quantitative analysis performed corresponds to flood quantile
sensitivity to

Annual maximum flood quantile variations for changes in

Relative changes in 10- and 100-year flood quantiles compared to
scenario 0 are evaluated for different situations, combining variations in

Figure 1 summarizes the results obtained and shows that changes in

Catchment production is highly influenced by the balance between rainfall
depth and the catchment storage capacity. Thus, sensitivity to the
production process should be analysed by introducing variability in rainfall
event depth for different

Arbitrary variations in

For each

Another point to be noted is the magnitude of relative variations depending
on the return period

Annual maximum flood quantile variations for scenarios 1.a
(

Climate scenarios considered for significance analysis.

Annual maximum flood quantile variations for scenarios
defined in Table 1 and

Confidence interval limits for a

The research presented herein highlights the filtering role brought on by catchment processes through a simple rainfall–runoff transfer function. The peak flow distribution is analytically derived from a rainfall model using the CN-SCS hydrological conceptualization. Variability of annual maximum peak flows is quantitatively analysed when changes in climatic input are introduced.

Such a modelling approach involves certain limitations, and yet it benefits from the analytical simplicity and practical applicability. Consequently, numerical results obtained after simulations cannot be transferred to hydrological regimes that differ from the type of Mediterranean catchments specified here. Nevertheless, the proposed methodology represents a useful modelling framework for further studies, and may constitute a first step forward towards a more complex analysis after relaxing some of the initial assumptions. Although certain dominant drivers of the hydrological response, like variability of watershed properties or land use changes, have not been explicitly considered in this study, the proposed modelling framework has the potential to incorporate those drivers to a certain extent, and thus, allow for the effect of such variability to be assessed and compared in future studies.

The results obtained from the sensitivity analysis can be summarized as
follows:

The filtering role of the catchment with regard to changes in the annual number of rainfall events is particularly significant when the rainfall event volume distribution is not strongly skewed.

Sensitivity to the runoff production parameters in the catchment is highly influenced by the balance between rainfall depth and catchment storage capacity. For higher return periods, relative changes in annual maximum flood quantiles tend to be asymptotically similar to those imposed by the climatic input. For low return periods, the infiltration process strongly influences the derived peak flow distribution, which is in accordance with typical Mediterranean catchment hydrological behaviour.

In the range of low return periods (1 to 10 years), the only parameter of the rainfall model which actually affects significantly peak flows is the mean rainfall event depth. The other parameters involved in the rainfall modelling approach play a negligible role in this case, mainly due to the threshold-based conceptualization used in the CN-SCS model.

The authors wish to thank Debra Westall for revising the paper. The present work was (partially) developed within the framework of the Panta Rhei Research Initiative of the International Association of Hydrological Sciences (IAHS). Edited by: S. Attinger