This paper proposes a methodology for estimating the transient probability
distribution of yearly hydrological variables conditional to an ensemble of
projections built from multiple general circulation models (GCMs), multiple
statistical downscaling methods (SDMs), and multiple hydrological models
(HMs). The methodology is based on the quasi-ergodic analysis of variance
(QE-ANOVA) framework that allows quantifying the contributions of the
different sources of total uncertainty, by critically taking account of
large-scale internal variability stemming from the transient evolution of
multiple GCM runs, and of small-scale internal variability derived from
multiple realizations of stochastic SDMs. This framework thus allows deriving
a hierarchy of climate and hydrological uncertainties, which depends on the
time horizon considered. It was initially developed for long-term climate
averages and is here extended jointly to (1) yearly anomalies and (2) low-flow
variables. It is applied to better understand possible transient futures
of both winter and summer low flows for two snow-influenced catchments in the
southern French Alps. The analysis takes advantage of a very large data set of
transient hydrological projections that combines in a comprehensive way 11
runs from four different GCMs, three SDMs with 10 stochastic realizations each, as
well as six diverse HMs. The change signal is a decrease in yearly low flows of
around

Incorporating global change in long-term water resource planning, water
management, and water governance is a major issue water managers currently
have to face

The first issue has been largely addressed in the literature over the last
decades, through the use of hydrometeorological modelling chains composed of
GCMs, downscaling techniques – either regional climate models or statistical
downscaling methods (SDMs) – and hydrological models (HMs). Such
hydrometeorological chains provide a quantification of the hydrological change
signal, as well as an estimate of the uncertainty associated to each level of
the modelling chain, provided of course that they include multiple models at
each level

These projections are however historically and still generally derived for
specific time slices in the future, and only few studies engaged in deriving
transient hydrological projections

The issue of quantifying internal climate variability and its additional
contribution to modelling uncertainty has retained much attention from the
climate community over the last few years

Lastly, the majority of hydrological change studies so far mainly focused on
uncertainties in the streamflow regime. Some of them explored changes in the
entire flow duration curve

The objective of this work is to deliver relevant information on possible
futures of low flows for informing water resource adaptation strategies. To
this aim, it attempts to answer the water manager's question by addressing
all four issues listed above for two specific snow-influenced Alpine
catchments with high stakes on water resources. This work takes advantage of
a very large data set of transient hydrological projections over the
1980–2065 period, that gathers all possible combinations of
hydrometeorological modelling chains built from 11 runs from four different
GCMs, three SDMs with 10 stochastic realizations each, as well as six diverse HMs.
Time series of mean annual minimum flow over 7 days are first derived
separately for winter and summer for both catchments and for each of the 1980
hydrological projections. The quasi-ergodic analysis of variance (QE-ANOVA)
framework developed by

Section

The Durance basin is located in the southern French Alps, and water flows
into the Rhône river. This basin has a total area of 14 000

Two case study catchments are considered here: the Durance@Serre-Ponçon
and the Verdon@Sainte-Croix (see Fig.

Delineation of the Durance basin and the two case
study catchments drawn on the gridded map of the 1980–2009 mean annual
precipitation from the SPAZM reanalysis

Reconstitutions of natural streamflow for both stations were provided by the
EDF power company which manages both Serre-Ponçon and Sainte-Croix
reservoirs. Reconstructed streamflow were derived prior to the R2D2-2050
project from outflows and stored volumes in the two reservoirs, and corrected
from the influence of other upstream hydropower reservoir operations. In this
study, these reconstructed streamflow time series were only used to calibrate
some of the HMs as detailed in Sect.

Climate projections over the Durance basin are based on global projections
from the ENSEMBLES project

Global model runs under the A1B emissions scenario.
Different runs from a given GCM correspond to simulations differing only from
their initial conditions

The spatial resolution of the global projections is not adapted to
hydrological modelling over small areas like the Durance basin. A downscaling
step has therefore been performed within a previous project on this basin

Statistical downscaling methods.

The archive for predictors is the NCEP/NCAR global reanalysis

The downscaling process thus led to 3300 (11 GCM runs

Six HMs have been run by different R2D2-2050 project partners over up to 26
catchments in the Durance basin during the project. Only simulations with
GCM-driven forcings described above at the two selected catchments described
in Sect.

HM characteristics. KGE refers to the Kling–Gupta efficiency

The hydrological modelling step thus led to 1980 transient hydrological projections – 330 downscaled climate projections multiplied by six HMs – for the period from 1 August 1958 to 31 July 2065. They include daily streamflow, actual evapotranspiration, and snow water equivalent for the two catchment case studies.

The low-flow indicator chosen here is the mean annual minimum flow over 7
days (MAM7)

Daily interannual regime of naturalized streamflow over the REF period for the two catchment case studies, and season boundaries for low-flow analysis. Grey ribbons frame the first and last deciles and the black line shows the median value.

The partitioning of uncertainties in hydrological projections is performed in
the framework of the quasi-ergodic analysis of variance (QE-ANOVA) framework
developed by

Previous applications of the QE-ANOVA framework focused on changes in
time-slice averages of the raw data

NFSs are estimated by first fitting trend models to the raw data

NFSs can be partitioned into GCM, SDM, and HM contributions through a
three-way ANOVA according to the following equation:

Winter low-flow NFS(

The internal climate variability variable

For the present ensemble of projections, estimates of both internal
variability components are derived with the quasi-ergodic assumption of
transient climate simulations for relative change variables, following
Appendix B of

Simple linear trend models are used to fit MAM7 projections of the whole
period considered (1980–2065), on the contrary to

A total of 72 linear trend models were fitted, one for each modelling chain, i.e. for
each combination of GCM, SDM and HM. The NFS of each modelling chain is then
obtained by considering relative changes with respect to the average of the
trend model for the associated chain over the 1980–2009 REF period following
Eq. (

Figure

In this study, the QE-ANOVA framework is extended for partitioning the
uncertainties not only on changes in time-slice averages as in the previous
applications but also on yearly anomalies of the raw values, in order to
capture the effects of year-to-year variability in the quantification of
uncertainties. The studied variable

The total variance and grand ensemble mean computed through the QE-ANOVA
approach allows deriving transient confidence bounds for the evolution of low
flows, provided that an assumption is made on the shape of the distribution.
Following previous uncertainty decomposition work on decadal averages, a
normal distribution is selected for 30-year low-flow averages

Having transient probabilistic projections further allows detecting the time
of emergence (ToE) of a change signal in low flows, i.e. the time when this
signal emerges from the underlying variability and uncertainty noise

The ToE analysis described above is also applied to perfect
hydrometeorological chains, i.e. chains with no GCM, SDM, or HM uncertainty.
The total variance is in this case estimated from internal variability
components and residuals only, and the grand ensemble mean is retained from
the analysis with actual modelling chains. Note that the latter assumption
requires adopting a truth-centred paradigm

A specific issue of interest in this study is the dependence of the low-flow
evolution on the HM used, all other things being equal. The fraction of
variance due to the HMs in the whole ensemble of hydrological projections as
given through the QE-ANOVA approach described above is checked against a
simple single-time ANOVA decomposition approach proposed by

Potential sources for the HM contribution to the total uncertainty are
further investigated through the evolution of selected HM state variables
potentially relevant for explaining the evolution of summer and/or winter low
flows. Computed summer low flows in snow-influenced catchments depend on two
main factors other than external meteorological forcings: evapotranspiration
and previous winter snowpack. More precisely, both

GCM effects on low-flow changes around the grand ensemble mean for both catchments and both seasons.

As for Fig.

As for Fig.

Fraction of total variance explained by each source of uncertainty for 30-year rolling averages of low-flow changes with respect to the REF period average. Values are plotted in the middle of each time slice.

Figure

Figure

Figure

The contribution of each source of uncertainty quantified by the QE-ANOVA
approach can be expressed as a fraction of the total variance for each lead
time

As for
Fig.

Projected changes in 30-year averages of low
flow for both stations and seasons, together with a partitioning of the
90 % confidence interval into the different uncertainty sources (see text
for details). Values are plotted in the middle of each time slice. The
fraction of the confidence interval for a given source of uncertainty is
proportional to the standard deviation of its contribution to the total
standard deviation, following

As for Fig.

Evolution of the probability of a low flow below the REF period average, for yearly anomalies and 30-year rolling averages, with the hydrometeorological model chains used here and with a perfect hydrometeorological model (see text for details).

Fraction of total uncertainty due to HMs computed from the QE-ANOVA and a simpler approach (see text for details), for both yearly anomalies and changes in 30-year rolling averages.

Relations between HM effects on low-flow anomaly and HM effects on AET/maxSWE anomaly for the year 2065. Significant relations at the 90 % confidence level are shown with solid lines.

Figure

Figure

Figure

Figure

Red curves in Fig.

Figure

Figure

The drivers of this high HM contribution to the total uncertainty – i.e. the
origin of effects from individual HMs shown in Fig.

The approach used here provides a transient evaluation of uncertainties in
yearly values or time-slice rolling averages in future low flows. It notably
allows to estimate the ToE of a decrease signal in low flows. The time series
approach at the heart of the QE-ANOVA framework makes this estimation rather
robust. This would not have been the case with other uncertainty estimation
approaches such as the one proposed by

More generally, all above statements rely on the assumptions of both the
QE-ANOVA framework described in Sect.

The HM contribution to total uncertainty shown in Fig.

Possible drivers of the divergence in HM responses has been explored through
the analysis of changes in model state variables AET and maxSWE.
Figure

A way forward to disentangle the origins of the divergence in low-flow
responses from different HMs in general would be to make use of the framework
for understanding structural errors

Finally, this study is based on the assumption that low-flow projections
derived from all individual HMs – but also from all individual GCMs and SDMs
– are equally valid. No simple relation could be found between present-day
performance in simulating interannual variability in low-flow anomalies and
HM effects. The robustness of the uncertainty decomposition results may
therefore be tested with subsets of HMs, as well as subsets of GCMs and SDMs.
It has to be noted that an experiment on HM uncertainty evolution following
removal of an outlier model has been recently performed by

The hydrological projection data set explored in this work includes a fairly
comprehensive list of uncertainty sources compared to most of previous
studies

The hierarchy of model uncertainties is, however, different from other
hydrological indicators. For changes in the mean annual streamflow of the
Durance catchment, SDM uncertainty was found to be larger than GCM
uncertainty

Additionally, some other potential sources of uncertainty were not considered. First, this data set is conditional on the single A1B emissions scenario, which should not be detrimental to results presented above given the relatively close time horizon considered. Adding the scenario uncertainty in the QE-ANOVA framework would be relatively straightforward, as it would take the form of an additional fixed effect alongside GCMs, SDMs, and HMs.

The uncertainty related to the temporal transferability of parameters –
whether from SDMs or HMs – has not been considered either in this study. The
hydrological uncertainty was found to be high when compared to that of SDMs
and GCMs, but it was also likely underestimated. Indeed, the time
transferability of HM parameters in a climate change context and its
contribution to overall uncertainties has recently been explored by some
studies

This paper proposes a methodology for estimating the transient probability
distribution of yearly hydrological variables conditional to an ensemble of
projections built from multiple GCMs, multiple SDMs, and multiple HMs.
The methodology is based on the QE-ANOVA framework that allows quantifying the contributions of the
different sources of total uncertainty, by critically taking account of
(1) large-scale internal variability stemming from the transient evolution of
multiple GCM runs, and (2) small-scale internal variability derived from
multiple realizations of stochastic SDMs. This framework thus allows deriving
a hierarchy of climate and hydrological uncertainties that depends on the
time horizon considered. It was initially developed for long-term climate
averages and is here extended to include year-to-year climate variability in
probabilistic hydrological projections, thereby following the recommendations
of

The QE-ANOVA framework is applied to better understand possible transient
futures of both winter and summer low flows for two snow-influenced
catchments in the southern French Alps. The analysis takes advantage of
a very large data set of daily transient hydrological projections over the
1981–2065 period that combines in a comprehensive way 11 runs from four
different GCMs, three SDMs with 10 stochastic realizations each, as well as six
diverse HMs. Results from the extended QE-ANOVA approach may be summarized
into three points. First, the change signal is a decrease in yearly low flows
of around

Two main conclusions can be drawn from the above analysis, leading to
corresponding lessons for future actions. First, internal variability brings
by far the largest part of the uncertainty in low flows for an individual
year in the future, even when the change signal is relatively large.
Increasing the robustness and resilience of water systems to future climate
conditions urges therefore water resources managers to first account for the
internal climate variability. The scientific focus should then be on
providing robust estimates of this variability by, for example, looking more
and further into the past to identify benchmark situations and events that
would serve as training sets for testing adaptation strategies, e.g. through
historical hydrometeorological reconstructions

Second, low-flow responses from different HMs diverge in a changing climate, presumably due to differences in both evapotranspiration and snowpack components resulting from the large range of approaches implemented in the six models used here. HMs should therefore be carefully checked for their robustness in a changed climate in order to increase the confidence in hydrological projections. In particular, efforts should made for validating the robustness of all components of HMs with specific analyses and relevant data sets, notably for evapotranspiration and snowpack evolution.

When a single GCM run is available for a given modelling chain

The variance in Eq. (

The large-scale internal variability component for any given chain

When multiple runs are available for a chain, this variance component is
estimated from all runs. The LSIV component of

Let

All authors collectively designed the experiments. Benoît Hingray and Jean-Philippe Vidal developed the model code and Jean-Philippe Vidal performed the simulations. Jean-Philippe Vidal prepared the manuscript with contributions from all co-authors.

This work was supported by funding from the French department in charge of
the environment through the R2D2-2050 project. Analyses were performed in