Hydrol. Earth Syst. Sci., 20, 4503–4524, 2016 www.hydrol-earth-syst-sci.net/20/4503/2016/ doi:10.5194/hess-20-4503-2016 © Author(s) 2016. CC Attribution 3.0 License. Climate elasticity of streamflow revisited – an elasticity index based on long-term hydrometeorological records Vazken Andréassian1, Laurent Coron1,a, Julien Lerat2, and Nicolas Le Moine3 1Irstea, Hydrosystems and Bioprocesses Research Unit (HBAN), Antony, France 2Bureau of Meteorology, Canberra, Australia 3Sorbonne Universités, UPMC Univ Paris 06, CNRS, EPHE, UMR 7619 Metis, Paris, France anow at: EDF-DTG, Toulouse, France Correspondence to: Vazken Andréassian ([email protected]) Received: 6 March 2015 – Published in Hydrol. Earth Syst. Sci. Discuss.: 1 April 2015 Revised: 12 July 2016 – Accepted: 23 August 2016 – Published: 10 November 2016 Abstract. We present a new method to derive the empiri- "Q=X, the elasticity of streamflow Q to a climate variable X, cal (i.e., data-based) elasticity of streamflow to precipitation is defined by the following equation: and potential evaporation. This method, which uses long- term hydrometeorological records, is tested on a set of 519 1Q=Q D "Q=X1X=X; (1) French catchments. We compare a total of five different ways to compute elas- where Q and X are the long-term average values of stream- ticity: the reference method first proposed by Sankarasub- flow and the climatic variable, respectively, and the opera- ramanian et al. (2001) and four alternatives differing in the tor 1 indicates the difference between the dated and average type of regression model chosen (OLS or GLS, univariate values. "Q=X is nondimensional (% = %), because it is a ra- or bivariate). We show that the bivariate GLS and OLS re- tio between two relative (and thus already nondimensional) gressions provide the most robust solution, because they ac- quantities. One can also define elasticity as the ratio between count for the co-variation of precipitation and potential evap- two absolute quantities and, provided both quantities are ex- −1 oration anomalies. We also compare empirical elasticity es- pressed in the same unit (for example, mmyr for stream- timates with theoretical estimates derived analytically from flow, precipitation or potential evaporation), it would still be −1 −1 the Turc–Mezentsev formula. a nondimensional ratio (mmyr = mmyr ). We will name Empirical elasticity offers a powerful means to test the ex- this absolute elasticity eQ=X, defined as trapolation capacity of those hydrological models that are to 1Q D e 1X: (2) be used to predict the impact of climatic changes. Q=X Table 1 summarizes the notations used in this paper. 1.2 Past studies on elasticity in hydrology 1 Introduction 1.2.1 Theoretical (model-based) studies 1.1 About hydrological elasticity Most of the studies on elasticity are theoretical, in the sense In a context of growing uncertainty regarding water resources that they are based on flows simulated by a hydrological due to climate change, simple tools able to provide robust model fed with different inputs. There are many examples estimates of this impact are essential to support policy and of such theoretical studies. Nemec and Schaake (1982) used planning decisions. Streamflow elasticity is one such tool: it the Sacramento model, Vogel et al. (1999) used the linear re- describes the sensitivity of the changes in streamflow related gression coefficients of annual streamflow models, Sankara- to changes in a climate variable (Schaake and Liu, 1989). subramanian et al. (2001) used the abcd model, Niemann and Published by Copernicus Publications on behalf of the European Geosciences Union. 4504 V. Andréassian et al.: Climate elasticity of streamflow revisited Table 1. Summary of the elasticity notations used in this paper (X being precipitation P or potential evaporation EP). Notation Definition Formula Relative streamflow elasticity – percent change of streamflow Q 1Q " D " 1X Q=X by percent change of climate variable X Q Q=X X Absolute streamflow elasticity – mm change of streamflow Q e 1Q D e · 1X Q=X by mm change of climate variable X Q=X Table 2. Comparison of the theoretical and empirical elasticity assessment methods. Theoretical (model-based) elasticity assessment Empirical (data-based) elasticity assessment Co-variations The modeling approach distinguishes between Problem: the changes in observed climatic variables of different the impact of different climatic variables can be correlated (e.g., 1P negatively correlated climatic (by keeping part of the forcing constant with 1T when the driest years are also the warmest), variables while modifying the other part). which makes it more difficult to attribute streamflow changes to one or the other variable. Data No need for long concomitant series of Long concomitant series of observed streamflow requirements observed streamflow and climatic variables and climatic variables are required. (only what is needed for model calibration). Extrapolation Extrapolates to extreme climatic changes Can only deal with the changes that capacity (i.e., to changes that have not been observed have been observed in the available over historical records). historical record. Eltahir (2005) used a purpose-built model and Chiew (2006) 1.2.3 Difference between theoretical (model-based) and used the SIMHYD and AWBM models. The most widely empirical (data-based) elasticity assessments used model in elasticity studies is the long-term water bal- ance formula first proposed by Turc and Mezentsev (Mezent- sev, 1955; Turc, 1954) (see Sect. 3.2). This formula (some- To clarify the differences existing between theoretical and times improperly confused with Budyko’s formula) was used empirical elasticity computing approaches, we have listed in elasticity studies by Dooge (1992), Arora (2002), Sankara- the key characteristics of both methods in Table 2. The subramanian et al. (2001), Yang et al. (2008), Potter and most important problem stems from the co-variation of po- Zhang (2009), Yang and Yang (2011), Donohue et al. (2011) tential evaporation (or temperature) and precipitation: Fu et and Yang et al. (2014), among others. al. (2007a) mentioned this issue and proposed to transform the “single parameter precipitation elasticity of streamflow 1.2.2 Empirical (data-based) studies index” into a “two parameter climate elasticity index” that would be a function of both precipitation and temperature, Only a few of the published elasticity studies are empir- in order to account for both effects simultaneously. Recently, ical. By empirical, we mean that they use measured data Chiew et al. (2013) underline that “because of the inverse (for different sub-periods) to evaluate the climate elastic- correlation between rainfall and temperature, any effect from ity of streamflow. To our knowledge, Sankarasubramanian the residual temperature on streamflow is much less apparent et al. (2001) were the first to publish a method based on than the direct effect of (the much more variable) rainfall”. the median of annual flow anomalies to compute elasticity, Note that the use of model simulations to compute stream- later used by Chiew (2006). Potter et al. (2010) analyzed flow elasticity circumvents this problem. concomitant reductions of precipitation and streamflow in However, there remains what we consider to be a major the Murray–Darling basin over three major historic droughts, disadvantage: since all hydrological models are a simplifica- and Potter et al. (2011) suggested computing elasticity as a tion of reality, using them to predict changes requires some multiple linear regression linking annual transformed stream- type of initial validation on empirical (observed) data. In- flow values to annual precipitation and temperature anoma- deed, we have recently compared (see Fig. 9a in Coron et lies. al., 2014) the ability of three models of increasing complex- ity to reproduce the variations in water balance equilibrium over 10-year long periods and shown that all three models tested had a tendency to underestimate observed changes. Hydrol. Earth Syst. Sci., 20, 4503–4524, 2016 www.hydrol-earth-syst-sci.net/20/4503/2016/ V. Andréassian et al.: Climate elasticity of streamflow revisited 4505 In this paper, we will focus on identifying the most ro- bust approach to computing empirical elasticity. Then we will compare the results obtained by this method with the theoretical elasticity of the Turc–Mezentsev water balance formula. This comparison will only aim at illustrating the dif- ference between the two approaches, since there is no reason to consider one or the other to be the “true” reference. 1.3 Scope of the paper In this paper, we test four alternative approaches to com- pute the empirical streamflow elasticity, which we compare over a large catchment set to the approach first suggested by Sankarasubramanian et al. (2001). In Sect. 2, we present the data set of 519 French catchments on which this study is based. Section 3 gives a short overview of the possible graph- ical representations of catchment elasticity and the methods used to quantify empirical elasticity. Section 4 presents a pre- liminary selection of the formulas, focusing on the distinc- Figure 1. Locations of the 519 French catchments analyzed in this tion between univariate and bivariate methods. Then Sect. 5 study. presents a regional analysis of streamflow elasticity to pre- cipitation and potential evaporation over France. Lastly, the conclusion identifies a few perspectives for further work. 2 Catchment data set Figure 1 presents the 519 catchments analyzed for these stud- ies. Long series of continuous daily streamflow and precipi- tation were available over the 1976–2006 period. The data set encompasses a variety of climatic conditions (oceanic, Mediterranean, continental, mountainous). Precipitation data were provided by Météo France as a gridded product, based on a countrywide interpolation of rain gage data (SAFRAN product; see Le Moigne, 2002). As far as potential evapora- tion data are concerned, we used the Penman–Shuttleworth equation (Shuttleworth, 1993) because Donohue et al. (2010) suggested that it was the most appropriate form of atmo- spheric evaporation demand when considering a changing climate.