The flow duration curve (FDC) is a fundamental signature of the hydrological cycle to support water management strategies. Despite many studies on this topic, its estimation in ungauged basins is still a relevant issue as the FDC is controlled by different types of processes at different time-space scales, thus resulting quite sensitive to the specific case study.

In this work, a regional spatially-smooth procedure to evaluate the annual
FDC in ungauged basins is proposed, based on the estimation of the

The proposed model is adapted to incorporate different “sub-models” to
account for local information within the regional framework, where
man-induced alterations are known, as common in non-pristine catchments. In
particular, we propose a module to consider the impact of existing/designed
water withdrawals on the

The procedure has been applied to a dataset of daily observation of about 120 gauged basins on the upper Po river basin in North-Western Italy.

Flow duration curves (FDC) are widely used tools to represent water availability in a river basin and are thus considered for many water resources planning and management purposes, like concessions for water uses and the planning of new hydropower plants. The FDC represents the percentage of time a certain value of discharge (usually at the daily scale) is equaled or exceeded in a river section over a specified a period of time. A FDC computed for a single year is called “annual”; if the observations of multiple years are merged together, the FDC is usually referred to as “period-of-record” or “total”. The empirical FDC can be easily built up by plotting the sorted the observations versus their frequency of non-exceedance computed with the Weibull plotting position, although the FDC is more frequently represented with respect to the exceedance frequency, thus resulting as a decreasing function. The evaluation of the FDC in ungauged basins is still a major issue in hydrological modeling, despite a large body of literature available on this topic, as recently reviewed by Castellarin et al. (2013).

In this work, we develop a regional model for prediction of the FDC in ungauged basins, developed and applied in the upper Po river basin (in North-Western Italy, an area mainly characterized by alpine and piedmont environments). The statistical framework developed is able to take advantage of local information about some types of anthropic effects. The aim of the work is to provide a regional tool to estimate the mean annual FDC in a generic watershed based on morphometric and climatic descriptors.

The regional spatially-smooth (SS) statistical estimation method proposed in
this paper is based on a framework developed by Laio et al. (2011) in the
context of regional flood frequency analysis. Analogies between the
procedures are obtained by representing the mean annual FDC by means of its

In the SS method the

The present approach is referred to as spatially-smooth because it does not
require the delineation of homogeneous regions, i.e. groups of basins sharing
the same statistical characteristics (and thus the same

Ultimately, this approach can be considered an extension of the index-flow approach, as the dimensionless FDC is not constant in a region but is allowed to change site by site.

To build the regional model, the sample

To provide reliable results, observed data should be natural flow observations, i.e. with no alterations due to upstream water uses. However, many gauging stations are located on rivers affected by man-induced alterations: in such cases, the observations should be previously processed to provide “naturalized” values. Where the actual amount of daily derived flow is known, corrections are straightforward, but this information is seldom available. This is the case (see Fig. 1) of run-on-river power plants where the gauging station is located between the intake and the outflow of the system and no information are available about the actual flow derived by the plant.

Example of gauging station bypassed by a water flow.

Observed and corrected

To overcome this problem, and exploiting also the data relative to such
stations, a new methodology, developed by Ganora et al. (2013), has been used
to obtain the

The correction method uses the parameter:

The available hydrological dataset refers to North-West of Italy (see Fig. 2) and includes 129 gauging stations with a total of 1438 station-year of daily observations, that have been selected after different quality controls. Only years with no more than 3 missing values have been included in the dataset. The available records have a mean length of 11 years, with actual durations ranging between 2 and 52 years. Many stations are quite recent and their data do not completely overlap with longer records, however they allow a larger spatial coverage of the model. More details about the dataset can be found in Ganora et al. (2013).

For each station, the annual FDCs have been computed based on the daily
discharge values; the

Location of gauging stations used in the analysis.

The database encompasses a number of basins affected by different water
withdrawals for different uses and with different intake systems. In many
cases, for our purposes, such anthropic effects can be neglected (e.g.,
negligible intake with respect to the average flow; high-elevation reservoir
affecting only the upper part of the river and with negligible effect on the
long-term statistics). In other cases alterations can be corrected with
measurements of real intake (e.g. large irrigation canals). However, other
cases require a correction of the observed streamflow statistics to provide
reliable “natural” values to be used in the regional analysis (with the
method described in the previous section). This is the case of 9 gauging
stations located downstream the intake of run-of-the-river hydropower plants
and upstream their water return. For these intakes no withdrawal data were
available, as no gauging recording system was installed. Table 1 reports the
main characteristics of the above mentioned stations with man-induced
alterations, as well as the corrected (“naturalized”)

Sample

Concerning the mean annual flow, to implement the regression model we
considered the transformed variable

Natural values of the

In order to select the most appropriate regional model, we performed all the
possible regressions combining 1 to 4 descriptors as independent variables to
estimate

Empirical (points) and analytical (lines) flow duration curve for
the Agogna at Momo river:

Concerning the

The final models, applicable to the whole area of interests are:

Performances of the selected regression models can be examined in Fig. 3
where predictions in cross-validation are reported. The mean annual discharge
prediction presents little scatter, while

The regional statistical framework presented is related to the estimation of
the

A first series of test of fitting was performed with the more common distributions (LN3, GLO, GPA, GAM). The main drawback observed was that in many cases negative streamflow values in the lower tail were obtained. This occurs when the distributions have a variable lower bound that can allow (depending on the parameter set) negative predictions for high exceedance frequencies. Further constraints to the parameters of the distribution can be added to keep the result consistent, but such controls seem hardly applicable in a regional context, for two reasons: first, the adjustments in general require numerical optimization that become difficult to be implemented in large-scale models; second, it is in general preferable to avoid complex (and likely unstable) estimation methods when the model will be used by practitioners that may not have confidence with numerical techniques.

To overcome the problem of negative quantiles without optimizing the
distribution's parameters, we introduce the use of the Burr probability
distribution, well known in different scientific communities but rarely used
in the hydrological field (Ganora and Laio, 2015). In particular, the
proposed function is the Burr XII distribution, originally proposed by
Burr (1942) as a two-parameter function, and later extended in the
three-parameter form (cumulative frequency function):

The Burr function results quite “flexible”, and thus suitable to fit many
different shapes of empirical data, thanks to the wide range of skewness and
kurtosis values it covers; nevertheless it has only three parameters, thus
avoiding overfitting problems. On the other hand, despite the simple
analytical form of the Burr XII, the estimation of the two shape parameters
is not straightforward as it requires the joint inversion of two nonlinear
equations (not shown here). For details about the estimation of the
parameters with the method of

To proceed with the model selection, a comparison between the different
distributions is in order. This comparison has been performed by fitting the
analytical curves to 365 quantiles corresponding to the non-exceedance
probabilities

Results show that LN3 and GPA have generally good fitting performances; GLO and B12 have slightly larger errors, but comparable with LN3 and GPA, while the GAM does not provide reliable results. However, the LN3 provides negative values in 21 % of the stations, the GLO in 51 % and the GPA in 11 %. The average duration (in days per year) affected by negative quantiles is 10, 14 and 19 days for the LN3, GLO and GPA respectively (the average is computed considering only the stations with at least one negative value). In the context of the data set analysed here the Burr distribution results the best choice, as it provides adequate fitting capabilities without the need of further optimization to avoid negative or inconsistent quantile predictions. An example of fitting is reported in Fig. 4a where the studied analytical functions have been superimposed on the empirical FDC.

Panel b of Fig. 4 shows the same record, but compared to the analytical FDCs
computed from the regional

A regional model for the estimation of the mean annual flow duration curve
(FDC) has been presented in the paper. The model is made of three
regression-based equations to estimate the

Moreover, the method allows the use in the calibration phase of records
affected by water derivations, even if the actual derived flow is not known
or hardly retrievable. This is possible thanks to a parsimonious corrections
method that provide “naturalized”

Finally, among different analytical representations of the FDC, the three-parameter Burr XII distribution has been proved to be the most suitable for the case study. In fact, while it provides good fitting performances, it does not allow the estimation of negative discharge values, which frequently appear by applying other analytical forms. This is advantageous as it is not necessary to optimize the parameters of a distribution to force the predicted quantiles to positive values only, thus making the model suitable for large-scale unsupervised applications.

Due to its flexibility, the present modeling framework is easily adaptable to
further extensions like, for instance, the inclusion of observations from new
gauging stations (by combining local and regional

Funding from the Italian Ministry of Education, Universities and Research (FIRB-RBFR12BA3Y project), and from the European Union – Horizon 2020 ERC-2014-CoG (“CWASI” project, ref. 647473) are acknowledged.