The methodology was applied to several basins located in three different states on the Gulf coast of the USA: Florida, Louisiana and Texas (Fig. 1). These basins were selected because they are characterized by a mild climate and, therefore, no snow events have been recorded, allowing us to neglect the snow melting effect. Daily streamflow discharge and precipitation values are available for each basin for different time windows (Table 1).

Daily streamflow discharge data were originally provided by the United States Geological Survey (USGS) gauges, while mean areal precipitation and climatic potential evaporation were supplied by the National Centers for Environmental Information, formerly the National Climate Data Center (NCDC) at daily resolution. The data set is a subset of the Model Parameter Estimation Experiment (MOPEX) database, used for hydrological model comparison studies (Duan et al., 2006) and for simultaneous calibration of hydrological models (Bárdossy et al., 2016).

2.1 Energy- and water-limited basins

Annual runoff variability is driven by the relative availability of water (i.e., precipitation) and energy (i.e., evaporation potential). Therefore, the weather is the most important driver of annual variability (Blöschl et al., 2013). Much of the annual runoff variability can be explained by observing the different availability of water and energy. For instance, if more water arrives at the basin than energy can remove through evaporation, the annual runoff will be high. Moreover, in this case the relationship between runoff and precipitation will be more linear than when more energy is available to evaporate the water. On the other hand, in an arid region, the aridity of the climate determines a high inter-annual runoff variability because of the nonlinear relationship between runoff and precipitation. Therefore, differences in water and energy availability cause differences in annual runoff variability. However, additional factors such as differences in seasonality and precipitation must be considered (Jothityangkoonad and Sivapalan, 2009). The relative availability of water and energy can be described through the Budyko curve (Budyko, 1974). The curve plots the ratio between mean annual actual evaporation and mean annual precipitation as a function of the ratio between mean annual potential evaporation and mean annual precipitation. Therefore, it defines a similarity index (i.e., the aridity index) to express the availability of water and energy and thus bolsters the classification of hydrological sceneries into various degrees of aridity. The Budyko curve represents the effects of water and energy availability on annual runoff variability. Moreover, it provides an indication of the synchrony of evaporation and precipitation. For instance, where precipitation and evaporation are in phase, runoff production declines since the basin allows for infiltration and stores water and vice versa. Many regions range from in phase to out of phase because of the strong seasonality of climate forcing. However, the climatic timing can influence runoff variability as presented by Montanari et al. (2006). In this framework, it is important to understand the behavior of the basins under analysis. To this end, we analyzed the mean annual runoff coefficient, the annual precipitation and the annual evapotranspiration against the annual mean temperature. This analysis is essential to understand the causal processes leading to the long-term mean and variability of runoff as also described in McMahon et al. (2013). The mean annual runoff coefficient is defined as

$$\begin{array}{}\text{(2)}& {\mathit{\mu }}_{\mathrm{R}}={\displaystyle \frac{\stackrel{\mathrm{‾}}{Q_{\mathrm{yr}}}}{\stackrel{\mathrm{‾}}{P_{\mathrm{yr}}}}},\end{array}$$

where $\stackrel{\mathrm{‾}}{Q_{\mathrm{yr}}}$ is the annual discharge volume and $\stackrel{\mathrm{‾}}{P_{\mathrm{yr}}}$ is the annual precipitation volume.

Results show that basins have two different behaviors: precipitation, evapotranspiration and runoff have either a positive or a negative correlation with the air temperature. In the former case the evapotranspiration is limited by the available water, which happens in water-limited basins; in the latter the evapotranspiration is limited by the available energy, which happens in energy-limited basins. For instance, measurements at Peace River (LA) suggest that the basin is balanced between energy and water limitation by the correlation criterion (Fig. 2 upper panel), while Ochlockonee River (FL), Amite River near Denham Springs (LA) and Bogue Chitto River (LA) are energy-limited. Results for Amite River are consistent with what was found by Carrillo et al. (2011). Since it is not possible to infer discharge values of a water-limited basin from the data set of an energy-limited one, analyses have been carried out on climatically homogeneous sets of basins.

Figure 2Annual precipitation against mean annual temperature (left panels), annual evapotranspiration against mean annual temperature (middle panels) and annual runoff coefficient against mean annual temperature (right panels) for four different basins: Peace River (FL), Ochlockonee River (FL), Amite River near Denham Springs (LA), and Bogue Chitto River (LA). In each plot, the Pearson correlation coefficient ρ is reported in the box.

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2.2 Preliminary analysis

The FDC can be interpreted as a distribution function of discharge over a given time period. To determine whether samples are drawn from the same distribution, here the two-sample Kolomogorov–Smirnov test (KS; Massey, 1951) is carried out on each pair of samples. The KS statistic on two samples is a non-parametric test for the null hypothesis that the two independent samples are drawn from the same continuous distribution. The decision to reject the null hypothesis is based on comparing the p value with the significance level set equal to 5 %. Moreover, the test allows us to estimate the distance between couples of FDC:

$$\begin{array}{}\text{(3)}& D^{*}=\left(\left|F_{\mathrm{1}}\left(x\right)-F_{\mathrm{2}}\left(x\right)\right|\right),\end{array}$$

where F₁(x) is the proportion of x₁ values less than or equal to x and F₂(x) is the proportion of x₂ values less than or equal to x. F₁ and F₂ are two FDCs. The KS statistic is applied to daily streamflow data sampled in several periods of record (e.g., 1 year, 10 years, 15 years). The long memory is relatively low, and we consider full years; thus, annual cycles do not have an influence on our results. The test is carried out both on pairs of samples gauged at the same location in two different years (or in two different decades) and on pairs sampled at two different sites. Since the streamflow data present autocorrelation, the autocorrelation affects the KS test. Weiss (1978) proposed a methodology to account for modifying the KS test for autocorrelated data. Later, Xu (2014) suggested a method that can be applied to a two-sample test. The information contained in the data is (usually) less than an independent and identically distributed (i.i.d.) sample with the same size. In other words, the number of equivalent independent observations is less than the sample size. In the following, we explain how we accounted for the equivalent sample size. It is easier to implement and, more importantly, it can be easily generalized to a two-sample test. We can assume that the autocorrelation effect attenuates after 3 d. For instance, let us take as an example a 1-year FDC. If the sample was 3 times smaller and for instance the length would equal 122 (i.e., 365 divided by 3), the null hypothesis would have been rejected anyway, leading to the same conclusion (i.e., the two samples cannot be regarded as the same distribution). This is due to the fact that, according to the two-sample KS test, the length of the equivalent sample that could pass the test should be 22.

The application of the KS test to our samples is pivotal to the development of the methodology. Test results show that streamflow data gauged in different periods (e.g., years or decades) at a specific location do not have the same distribution. The consequence is that it is not possible to use the parameters and the distribution derived from a FDC built for a specific time window to build the FDC of another time window. The same results comparing streamflow data were gauged in a specific year or decade at two different sites. Since the two data sets cannot be regarded as the same distribution, it is not possible to derive the FDC at one location using the parameters of the FDC sampled at another location. Therefore, it is necessary to develop a methodology that accounts for the weather, as it is the main driver of FDC variability as shown in the following. Figure 3 shows the magnitude of the difference between FDCs built at the same location using streamflow data gauged during different time windows.

Figure 3FDCs built for Tangipahoa River (FL) for four different hydrological years. Every hydrological year starts in October and ends the following September.

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The aim of this paper is to find the distribution of Q_k(t) for a time period (T₁, T₂), that is, a FDC. We assume that discharge is related to precipitation in the form

where k is a generic site, h_k is the transformation, usually approximated by a hydrological model, P_k is the precipitation and β_k is the specific parameter of the hydrological model. The core of this work is to retrieve the discharge values without hydrological modeling as modeling often introduces additional errors and may be biased for long subperiods. Thus, the main idea is to get rid of a complicated nonlinear process and to find a filter which relates the distributions.

The main hypothesis underlying this work is that daily flow duration curves at a partially ungauged location can be found with knowledge of the precipitation record at a donor site. The most important descriptor of the weather characteristic is the rainfall; however, we cannot use the distribution of P_k to assess the FDC directly as it will fail due to the lacking temporal structure and the many zeros. We can then use a transformation of P_k, the Antecedent Precipitation Index (API):

Both transformations reported in Eqs. (4) and (5) can be regarded as filters acting on P_k. These filters do not necessarily produce highly correlated series, but may produce series with similar distributions. The API is used to investigate precipitation data in a similar way to discharge data as it combines in a streamflow-like way the history of the precipitation. It represents the memory of a basin as it is related to the amount of water released by the soil to the river considering a given time window. Specifically, the API allows us to take into account the antecedent conditions and the duration of the rainfall events and gives an estimate of the portion of rainfall contributing to storm runoff (Linsley et al., 1949). It is a sequence of linear combinations of rainfall events in the period preceding a specific storm (Kohler and Linsley, 1951). For a resolution of 1 d and a time window of 30 d, the API at the ith day is given by

where α is a constant and ranges from 0 to 1 and P_i is the daily precipitation that occurred on the ith day (Kohler and Linsley, 1951). When α tends to zero, API keeps tracks of the precipitation that occurred on the few previous days, and it represents the short memory of the basin. When α tends to 1, the API represents the long memory of the basin as it includes the effect of precipitation that occurred many days before. To capture the latter behavior, in this study α is chosen equal to 0.85. This is in agreement with a previous study by Sugimoto et al. (2016), who investigated a case study area whereby a preliminary analysis was performed (i.e., Neckar basin); nevertheless, this value was found to be suitable also for the US basins. Here the API is calculated from areal precipitation instead of point precipitation.

Formally the basic hypotheses of this paper are the following.

• Flow duration curves are not invariant properties of basins, but are the product of basin, weather and human interactions. In this investigation we do not consider the human interactions.

• Precipitation is the most important influencing factor on discharge.

• Basins delay the reaction on precipitation; therefore, the API is a better indicator of the influence of precipitation on discharge.

• We assume that discharge and API are changing in a similar way for longer time periods.

Let $F_{\mathrm{A},T_i}\left(q\right)$ be the distribution of daily discharges at basin A and time period T_i (flow duration curve for the selected time period) and $G_{\mathrm{A},T_i}\left(a\right)$ be the distribution of daily API at basin A and time period T_i.

The transformation from T_i to T_j provides an estimated $F_{\mathrm{A},T_j}^{*}\left(q\right)$:

This is a quantile–quantile transformation.

The basic question can be written in the form of the following equation:

That can be summarized in the following question: do the percentiles of the API change in the same way as those of the discharge? Note that if the relationship between API and discharge is a good one-to-one relationship, then the two sides are nearly equal. Even a weak relationship can do a good job if the errors are independent and the signal of the change is correct. Figures 4 and 5 show the difference between the real change in percentiles and that obtained by using the API for different time periods according to Eq. (9). Note that the assumption that the FDC is time-invariant would imply that the lines for the discharge are on the diagonal.

Figure 4The transformation functions of the 1951–1960 FDC (solid) and API (dashed) to the target periods 1971–1980 (blue) and 1981–1990 (orange) for Amite. For the sake of comparison, the diagonal is dotted in orange.

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Figure 5The transformation functions of the 1951–1960 FDC (solid) and API (dashed) to the target periods 1971–1980 (blue) and 1981–1990 (orange) for Bogue. For the sake of comparison, the diagonal is dotted in orange.

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The correlation between API and discharge is around 0.6, but the transformations are quite similar and the API-based transformation delivers good FDCs.

If the API is changing continuously in space, then one can use the change in the FDC of a different location B for the estimation:

In the following, the methodology is reported step by step; then, the performance criteria used to estimate the goodness of the methodology are presented.

Figure 6Illustration of FDC generation using the interpolation with the API of the donor site as a proxy.

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The procedure explained above was tested on several target basins varying both donor and target periods.

Figure 7Interpolated FDC at Tangipahoa River (FL) and Bogue River (LA), in the upper and lower panels, respectively. The donor basin is Blanco River (TX). The donor years are a 20-year time window from October 1948 to September 1968 and from October 1968 to September 1988. Target years are the decades shown above each panel. Blue and red dots are the observed and interpolated FDC at the target basin during the target period, respectively; the black dots are the observed FDC at the donor basin during the target period. In each box the Kolmogorov–Smirnov distance between observed and interpolated values, D^*, is reported. The p value of the test is always around zero, but for Tangipahoa target years 1948–1958.

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Results show a good agreement between observed and interpolated FDCs. For instance, the FDCs interpolated using 20 and 10 years as donor and target periods, respectively, have a good performance, as shown for Tangipahoa and Bogue basins (Fig. 7). The method performance is higher for intermediate durations, while it can be lower for the low flows, e.g., as at Bogue for target years 1988–1998 (Fig. 7, lower panels) and for the high flows. The good performance of the approach is also noticeable when the target period is 15 years (Fig. 8). In each panel, the two-sample Kolmogorov–Smirnov test distance between observed and interpolated values, D^*, is reported. D^* is characterized by small values showing a good performance of the method. Since usually the FDC of a donor site is used to retrieve the FDC of a target site for the same period, the FDC of the donor basin recorded during the target period is also plotted. It is noteworthy to observe that the difference between these two FDCs can be substantial. This implies that the FDCs can be substantially different at different sites in the same period of time.

Figure 8Interpolated FDC at Tangipahoa River (FL), Bogue River (LA) and Choctawhatchee River (FL), in the upper, middle and lower panels, respectively. The donor basin is Blanco River (TX). The donor and target years are periods of 15 years. The blue and red dots are observed and interpolated FDC, respectively, at the target basin during the target period; the black dots are the observed FDC at the donor basin during the target period. In each box the KS distance, D^*, is reported. The p value of D^* is always around zero.

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Interpolated and observed FDCs almost perfectly match when obtained using long donor and target periods (Figs. 7 and 8). On the other hand, when the target period is short, the performance decreases, as also shown by the KS distance, D^*, reported in each single panel of Fig. 9 where the target period equals 1 year. As a matter of fact, the donor period being constant, the KS distance is much higher when the target period is 1 year (Fig. 9) and the p value of the test is always zero, but for hydrologic years 1972–1973 and 1976–1977. Nevertheless, the interpolated and observed FDCs have a high agreement in shape, as for instance at Tangipahoa River for all but one (i.e., 1969–1970) target year. In these cases, the difference between the two curves could be due to the different temperature values characterizing the donor and target basins. This affects the evapotranspiration in the two basins and, therefore, the streamflow values.

Figure 9Interpolated FDC at Tangipahoa River (FL). The donor basin is Blanco River (TX). The donor years are a 20-year time window from October 1948 to September 1968. Target years are each hydrological year from October 1968 to September 1983. The blue and red dots are observed and interpolated FDC, respectively, at the target basin during the target period; the black dots are the observed FDC at the donor basin during the target period. In each box the Kolmogorov–Smirnov distance, D^*, is reported. The p value of the test is always around zero, but for hydrologic years 1972–1973 and 1976–1977.

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Results suggest that the API gives effectively a good estimation of the memory of the basin and can be used to represent the precipitation similarly to the discharge.

To estimate the goodness of the methodology, the NSE, BIAS and MAE are evaluated for the 1st, 3rd, 5th, 10th, 20th, 30th, 50th, 75th, 90th and 99th percentiles.

When a decade is used as both target and donor period, the performance measures show a good agreement between observed and interpolated values (Fig. 10). The NSE index shows accurate estimation; i.e., it is characterized by values close to 1, especially of intermediate percentiles. The BIAS provides information regarding the overall agreement between interpolated and observed values. Its magnitude is likely higher for high flows, while it attenuates for intermediate percentiles. The MAE also shows a low performance for high streamflow values. This is due to the fact that the procedure is more able to reproduce the average streamflow values than extreme events such as high and low flows. However, low flows are more likely well estimated rather than high flows.

Figure 10Performance measures NSE, BIAS and MAE evaluated for specific percentiles (on the y axis) and for specific target decades on the x axis. The donor decade is 1948–1958 and the donor basin is Blanco (TX). Each target basin is indicated in the corresponding box. Negative values of the NSE as well as outliers of BIAS and MAE are reported in the corresponding box.

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When both target and donor periods equal 15 years, the agreement between interpolated and observed flow values is high (Fig. 11). The NSE shows values of efficiency around 1; thus, there is a good match between interpolated and observed values, even though there are a few exceptions. The errors are very low in value, as shown by the MAE, which also reveals a poor performance for high flows, while the performance improves for intermediate and low flows. The high flows are more likely estimated with a higher error than intermediate and low flows, as also shown by the BIAS.

Figure 11Performance measures NSE, BIAS and MAE evaluated for specific percentiles (on the y axis) and for specific 15 target years (i.e., 1963–1978 and 1978–1993 on the x axis). The donor decade is 1948–1963 and the donor basin is Blanco (TX). Each target basin is indicated in the corresponding box. Negative values of the NSE are reported in the corresponding box.

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As resulted from the KS test applied to pairs of FDCs obtained from recorded data at the same site in different periods, FDCs cannot be considered an invariant characteristic of a basin. The fact that FDCs are not invariant suggests that the weather is a driver of annual runoff variability. Indeed, the reason should be found in the weather conditions, as others (e.g., the basin area, the land use) did not change. To better investigate these findings, we performed the KS test on pairs of observed and interpolated FDCs for two purposes. The first is to know whether pairs of interpolated and observed FDCs at the same site have the same continuous distribution; the second is to know which is the distance between these pairs. The test performed on pairs of interpolated and observed FDCs revealed that the null hypothesis could not be rejected for nearly half of the cases. For instance, for Tangipahoa River the test was not rejected in 48 % of the cases (Fig. 12b). On the contrary, the test rejected the null hypothesis that FDCs built at the same location in different periods had the same distribution. In 73 % of the cases, the distance between pairs of interpolated and observed FDCs of the same period is smaller than the distance between FDCs built at the same site from data recorded during different periods (Fig. 12b and a, respectively). These results suggest that the methodology proposed here has a good performance, and it is actually an interesting alternative to other methodologies, which assume that FDCs of different periods have the same distribution.

Figure 12Kolmogorov–Smirnov distance between couples of streamflow values observed (a) and between couples of streamflow values observed and interpolated (b) at Tangipahoa River (FL) from October 1948 to September 1987.

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As the weather conditions strongly influence the FDC estimation, we analyzed the streamflow percentiles to assess the between-year variability. To this end, the moving average (MA) of the 30th, 70th, 90th and 95th percentiles of streamflow is estimated. The MA values are estimated using three different fixed time windows (i.e., 10, 15 and 20 years; Fig. 13).

Figure 13Moving average (MA) of the 30th, 70th, 90th and 95th percentiles of daily streamflow values gauged at Tangipahoa. Three different fixed time windows are used to estimate the MA: 10, 15 and 20 years. On the x axis the first year of each interval is plotted (Y^*).

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It is interesting to observe that the MA values are characterized by a strong variability throughout the time. The fluctuation of the flow percentiles suggests that the percentiles cannot be considered an invariant characteristic of the basin. Therefore, it is not possible to estimate the flow quantiles using regression methods that do not consider the weather characteristics. These methods, first, regionalize empirical runoff percentiles using multiple regression models. Then, regional evaluations of flow percentiles are interpolated across the percentiles (e.g., Franchini and Suppo, 1996; Smakhtin, 2001). If flow percentiles are estimated separately from weather characteristics, it may result in a misrepresentation of the percentiles themselves. Therefore, we suggest adding a weather factor to account for the influence of the weather in the percentile estimates.

The streamflow is obtained from USGS National Water Information System (NWIS), available at http://water.usgs.gov/nwis (last access: 15 April 2020) (USGS, 2020). Mean areal precipitation and climatic potential evaporation were supplied by the National Centers for Environmental Information, formerly the National Climate Data Center (NCDC), at daily resolution (https://www.ncdc.noaa.gov/data-access, last access: 15 April 2020) (NOAA, 2020).

AB and ER jointly developed the concept and the methodology of the study. HK and ER performed the computations and the analysis for the case study. ER wrote the first draft of the manuscript. The manuscript was revised by ER and AB.

The authors declare that they have no conflict of interest.

Elena Ridolfi has been supported by the Centre of Natural Hazards and Disaster Science (CNDS) in Sweden (https://www.cnds.se/, last access: 15 April 2020).

This paper was edited by Stacey Archfield and reviewed by William Farmer and Thomas Over.