Research papersThe analytical derivation of multiple elasticities of runoff to climate change and catchment characteristics alteration
Introduction
Climate change was widely reported in the past decades and is expected to continue (IPCC, 2007). Concurrently, human activities increased due to strong growths in population, including cultivation, irrigation, afforestation, deforestation and urban construction (Brown et al., 2005, Scanlon et al., 2007, Roderick and Farquhar, 2011). Both climate change and increasing human activities intensify the environmental changes that are occurring around the world. Understanding the interactions between climate change, human-induced land use/cover change and the terrestrial water cycle under the changing environment is critical for many research disciplines, such as hydrology, meteorology and ecology (Eagleson, 2002, Rodriguez-Iturbe and Porporato, 2004, Li et al., 2013). Water resources planning and management urgently need to incorporate climate variability and anthropogenic activities to develop sustainable strategies. Therefore, a deceptively simple question - how do climate variability and human activities respectively attribute to the change of annual runoff? - has attracted the attention of hydrologists and water resources managers and planners (e.g., Lane et al., 2005, Siriwardena et al., 2006, Tuteja et al., 2007, St Jacques et al., 2010, Wang et al., 2013a). Traditionally, paired catchment method and physically based method are used to quantitatively estimate the contributions of climate change and human activities to changes in the water cycle. The paired catchment method is considered to be the foundation of studying forestry and vegetation effects on hydrology (Brown et al., 2005). However, paired catchments are not always available due to difficulties in locating suitable controls and are expensive to setup (Fohrer et al., 2005, Wei and Zhang, 2010, Zhao et al., 2010). The physically based approaches (by the means of distributed physically-based hydrological models) are always limited due to the time-consuming model calibration, the requirement of large data sets and high uncertainties in the model structure and parameter estimation (Zhao et al., 2010, Wang et al., 2013a, Xu et al., 2014).
Recently, an approach based on climate elasticity, which was first introduced by Schaake (1990), is being developed actively to assess the climatic and anthropogenic effects on runoff due to its clear physical meanings and simple formulation (e.g., Dooge, 1992, Sankarasubramanian et al., 2001, Fu et al., 2007, Zheng et al., 2009, Yang and Yang, 2011, Roderick and Farquhar, 2011). The climate elasticity of runoff is defined as the proportional change in runoff due to the proportional change in one or more climatic variables (Sankarasubramanian et al., 2001, Fu et al., 2007). During the past two decades, the climate elasticity of runoff has roughly undergone three major expansions. Schaake (1990) defined the precipitation (P) elasticity to runoff (Q) as:and the impacts of P on Q can be evaluated by:
Based on Eq. (2), a non-parametric approach was introduced by Sankarasubramanian et al. (2001) and found to be robust via Monte Carlo experiments in three basins of the United States when directly identifying the response of runoff to precipitation based on observed long-term hydrometeorological data. Many studies have employed Eq. (2) to investigate the impact of precipitation change on annual runoff (e.g., Sankarasubramanian and Vogel, 2003, Niemann and Eltahir, 2005, Chiew, 2006, Novotny and Stefan, 2007).
Apart from precipitation change, air-temperature change can also have important impacts on runoff, which is particularly true under global warming. To simultaneously assess the impact of precipitation and temperature on runoff, Fu et al. (2007) extended Eq. (2) to consider both precipitation and temperature elasticities by:where and are the P elasticity and air temperature (T) elasticity, respectively, and revealed the critical role of T in regional runoff generation in two typical rivers: the Spokane River Basin in the USA and the Yellow River Basin in China. Fu et al. (2009) further revealed the potentially serious changes in water resources management in the North China Plain under future climate scenarios based on this equation. Obviously, in terms of climate change, integrating the effects of temperature, wind speed, solar radiation and vapor pressure, potential evapotranspiration is more competent than considering only temperature when investigating the influence of climate on hydrological processes, particularly in the case of the widely reported “evaporation paradox” in recent years. Zheng et al. (2009) afterwards proposed a similar expression of climate elasticity of runoff but in terms of precipitation and potential evapotranspiration as:where and are the P elasticity and potential evapotranspiration (E0) elasticity, respectively, and applied it to quantify the elasticity of runoff to climate and catchment characteristics change.
Note that potential evapotranspiration is not directly measured, and temperature is not the only climate variable that affects potential evapotranspiration. Yang and Yang (2011) further evaluated the effects of more climatic variables such as the net radiation and wind speed on runoff by expanding E0 in terms of temperature, net radiation (Rn), wind speed (U) at a height of 2 m above the ground and relative humidity (RH), producing a new equation:where , , , and are the elasticities of runoff with respect to P, T, , U and RH respectively. The spatial variations of climate elasticities across China were further characterized in Yang et al. (2014a). Based on this equation, Xu et al. (2014) conducted an attribution analysis to quantify the impacts of climate variation and land use/cover change on the decline of runoff in the Hai River Basin. The major expansions of climate elasticity of runoff and their applications have been summarized in Table 1.
In addition to climate change, the alteration of catchment characteristics can also play a key role in the variability of the flow regime and subsequently the amount of available water resources, particularly in rapidly developing regions, although catchment characteristics were always assumed to remain constant over time in climate-change impact studies. In this study, the authors summary that the Budyko framework with two climatic variables and one catchment-specific parameter, describing the dependence of actual evapotranspiration on energy availability and water availability, made it possible to include the effect of changes in catchment characteristics over time when employing the analytical model to construct elasticities. Roderick and Farquhar (2011) appears to have been the first to present analytical expressions to qualitatively estimate the impact of catchment characteristic alteration and climatic change on runoff change within the Budyko framework. Recently, Liang et al. (2015) quantify the impacts of climate change and ecological restoration on runoff changes in the Loess Plateau of China based on this framework.
With long-standing interest in theoretical aspects of the “Budyko curve”, researchers successively proposed more analytical equations during past decades. Among these equations, two threads of independent theoretical development have attracted many attentions (Roderick and Farquhar, 2011), originating respectively from Russian scholars (Bagrov, 1953, Mezentsev, 1955) in the form of(for the generalized form of Eq. (6), see Choudhury, 1999) and Chinese scholar (Fu, 1981) in the form of(for the generalized form of Eq. (7), see Zhang et al., 2004), where E is the actual evapotranspiration; and and are the empirical dimensionless parameters, respectively, which play similar roles in their equations to determine the shape of the Budyko curve and reflect the impact of catchment characteristics. Recently, with the theoretically derived Choudhury’s equation (Eq. (6)) based on Fu’s approach, these two equations were reconciled by Yang et al. (2008). Further analysis using the data from numerous catchments in China indicated that the catchment parameters in the two equations are more and less linearly related through (Sun, 2007, Yang et al., 2008).
In practice, some issues still need to be addressed in the development of the climatic elasticity, particularly for the impact assessment study. To date, four evaporation-related climatic variables were usually considered in the study of the climatic elasticity of runoff: mean air temperature, radiation, wind speed and relative humidity (see Eq. (5)). However, for the purpose of impact assessment study and with the consideration of data availability, the following issues need to be considered. (1) As the primary representation of climate change, warming trends are actually proved to be more pronounced in minimum temperature (night time temperature) (Alexander et al., 2006, Wang et al., 2013b). Compared to the average air temperature, the extreme temperatures are of more significance to climate change and thus should be considered as key variables when the climate elasticity is used to assess climate change effects on runoff. (2) In FAO Penman-Montieth equation, net radiation (Rn) is computed from sunshine duration (N), relative humidity (RH), maximum temperature (TA) and minimum temperature (TI) and thus correlates with them. Such correlations may cause relative errors when taking Rn, RH, TA and TI as the key variables to estimate the climatic effect on runoff. Furthermore, limited by the cost of measurement equipment and the difficulties of its maintenance and calibration (Hunt et al., 1998, Thornton and Running, 1999, Chen and Li, 2013), the accurate long-term records of solar radiation are not widely available. Therefore, the direct use of radiation in climate-change impact studies might not be widely applicable in practice. (3) In recent studies on the runoff change attribution analysis with elasticity method (e.g., Yang and Yang, 2011, Meng and Mo, 2012), although the effects of evaporation-related climatic variables (e.g., air temperature, wind speed, relative humidity and radiation) on runoff were evaluated and separated, the effect of alteration of catchment characteristics on runoff change was ignored, that is, the interannual changes in catchment characteristics was not considered. Therefore, an improved multiple climate elasticity as well as catchment characteristics elasticity of runoff need to be derived with the consideration of the influence of correlation and its applicability.
From the aspect of model complexity, the performances of different models with different complexity/extension have not been investigated, and the requirements of these models have not been clearly delineated. Although it is not clear how many climatic variables should be theoretically considered in the assessment of the climatic effect on runoff, it is adequate to use the precipitation and evapotranspiration elasticities (see the Eq. (4)) to describe the complex non-linear relationship between the hydrological and climatic regimes based on the normal water balance theory if one only wants to identify the integrated effect of climate change. More importantly, from the mathematical perspective for a given response variable, more assessment errors may be produced if more control variables are used (i.e., more elasticities are considered). This may produce a dilemma regarding additional insights on the effect of climate in term of more climate variables versus possibly more reliable evaluation.
To address these issues, we extend five climatic elasticities into six parameters elasticities by using the widely recommended Penman-Montieth equation in deriving reference evapotranspiration to fully consider the maximum and minimum temperature effect and alleviate its influence on the assessment results from the correlation among climatic variables. That is, the climate elasticity of runoff is considered to be a function of precipitation, sunshine duration, wind speed, relative humidity, minimum and maximum temperature. An analytical framework based on the Budyko hypothesis was subsequently established to decompose the climate-induced and catchment characteristics-induced runoff change. As a case study, the method was applied to 30 hydrological stations in major rivers across China. Moreover, as a preliminary study, we also assessed the applicability and reliability of climate elasticities with different numbers of climate factors by investigating their performances on the quantitative attributions to runoff. This paper is organized as follows. The derivation of multiple elasticity is given in the next section, followed by the results of the analysis and application in 30 catchments. The discussion, including the necessity of multiple elasticities, the discussion on the role of catchment property in runoff change and the uncertainties analysis, are presented before the conclusions.
Section snippets
Elasticity of runoff derived from the Budyko hypothesis equation
The water balance for a catchment can be described by . On the annual scale, the transient changes in soil water storage () can reasonably be neglected with the assumption that the change in in a given catchment is from one steady state to another steady state, and can thus be considered equal to zero for a long period. Thus, the normal water balance equation can be simplified as , leading to the differential form:
Eq. (6) can be written as to estimate the
Study area and data processing
The elasticity theory and decomposition framework formulated in this study is used to quantitatively assess the effects of the climate and catchment characteristics on runoff in the primary, large river basins across China including the Songhua River Basin, the Liao River Basin, the Hai River Basin, the Yellow River Basin, the Huai River Basin, the Yangtze River Basin, and the Pearl River Basin. The monthly runoff data from 30 hydrological stations in these seven major basins were collected
Long-term trends analysis and breakpoint detection for a hydrometeorological series
Table 3 shows the statistical results of the long-term trends and their significance for hydrometeorological variables based on the nonparametric Mann-Kendall test (Mann, 1945, Kendall, 1975) from 1956 to 2000. Significantly positive trends in can also be found in the seven basins (25 catchments for total 30 catchments). However, only a few stations had significantly positive trends in . Therefore, the warming processes in the study area were primarily characterized by increasing minimum
The necessity and applicability of multiple elasticities of runoff
Elasticity can well represent the impact of a given change on the normal state of a system (Harman et al., 2011) that is readily calculable and thus has been widely used to assess a hydrological regime with respect to environmental changes in past decades. From 1990, the runoff elasticity has undergone the development process from a single-parameter (e.g., precipitation) climatic elasticity (e.g., Schaake, 1990) to a two-parameter (e.g., precipitation and temperature) climate elasticity (e.g.,
Conclusion
To reflect the effect of maximum and minimum temperature on runoff and avoid the errors from the correlation between radiation, air temperature and relative humidity, a five evaporation-related variables analytical derivation of climate elasticities was proposed in this study to further separate and evaluate the climatic effect on runoff more systematically. Based on this derivation, a decomposition framework with the ability to independently estimate the contributions of climatic and
Acknowledgments
This work was jointly supported by the National Natural Science Foundation of China (51379057), the Fundamental Research Funds for the Central Universities (2015B14114), National “Ten Thousand Program” Youth Talent, the Australian Endeavour Research Fellowship, CSIRO Computational and Simulation Sciences TCP, and QingLan Project. Cordial thanks are extended to the Editor, Professor Geoff Syme, and two anonymous referees for their valuable comments.
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