Journal of Hydrology 522 (2015) 326–338

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Separating the impacts of climate change and human activities on runoffusing the Budyko-type equations with time-varying parameters

http://dx.doi.org/10.1016/j.jhydrol.2014.12.0600022-1694/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +86 13871078660; fax: +86 27 68773568.E-mail addresses: jc1207@whu.edu.cn (C. Jiang), xionglh@whu.edu.cn (L. Xiong),

dingbao.wang@ucf.edu (D. Wang), liupan@whu.edu.cn (P. Liu), slguo@whu.edu.cn(S. Guo), c.y.xu@geo.uio.no (C.-Y. Xu).

Cong Jiang a, Lihua Xiong a,⇑, Dingbao Wang b, Pan Liu a, Shenglian Guo a, Chong-Yu Xu a,c

a State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, Chinab Department of Civil, Environmental, and Construction Engineering, University of Central Florida, Orlando, FL, USAc Department of Geosciences, University of Oslo, P.O. Box 1047 Blindern, N-0316 Oslo, Norway

a r t i c l e i n f o

Article history:Received 15 September 2014Received in revised form 23 December 2014Accepted 29 December 2014Available online 7 January 2015This manuscript was handled byKonstantine P. Georgakakos, Editor-in-Chief,with the assistance of Matthew Rodell,Associate Editor

Keywords:Runoff changeBudyko-type equationTime-varying parameterMoving windowClimate changeHuman activities

s u m m a r y

The Budyko-type equations have begun to be widely adopted to separate the contributions of climatechange and human activities to the variation of runoff over long-term periods by using the multi-yearaverages of hydrological variables. In this study, a two-step framework based on four single-parameterBudyko-type equations is proposed to separate the impacts of climate change and human activities onrunoff. First, the relationship of the parameter w in each Budyko-type equation with climatic and humanfactors is built to reveal the time-variation process of w by using an 11-year moving window. Second, theimpacts of climate change and human activities on runoff are separated by using both the decompositionmethod and the sensitivity method. This separating framework is applied to analyze the variation of therunoff during 1960–2009 in the Weihe River. It is found that the parameter w in each Budyko-type equa-tion is significantly related to both factors of climate and human activities. The results from both thedecomposition method and the sensitivity method show that climate change is the main driving factorto the decline in runoff of the Weihe River, while human activities are another important factor. In gen-eral, climate change affects runoff not only by changing the hydrological inputs (precipitation and poten-tial evaporation) but also by altering the watershed characteristics as represented by the parameter w;while the impacts of human activities on runoff are exerted mainly through the alteration of thewatershed characteristics.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

Plenty of studies have revealed that runoff for many rivers allover the world has presented variations at various time scalesdue to climate change and/or human activities (McCabe andWolock, 2002; Kahya and Kalayci, 2004; Birsan et al., 2005;Chiew and McMahon, 2006; Khaliq et al., 2009; Kumar et al.,2009; Velpuri and Senay, 2013). These variations in runoff havebeen an urgent challenge for water resources planning and man-agement (Milly et al., 2008). In response to this challenge, one ofthe tasks for researchers is to identify the different roles of climatechange and human activities on the nonstationarities in runoff.

Both deterministic rainfall–runoff models and statistical meth-ods have been proposed to assess the impacts of climate change orhuman activities on runoff (e.g., Gosling, in press; Liu et al., 2013;

Zhou et al., 2013). For example, Chiew et al. (2009) used a concep-tual rainfall–runoff model SIMHYD to estimate climate changeimpact on runoff across southeast Australia. Ma et al. (2010)applied a distributed hydrological model to assess the impacts ofclimate variability and human activities on runoff decrease in theMiyun Reservoir catchment located in North China. Mango et al.(2011) used the Soil Water Assessment Tool (SWAT) to investigateland use and climate change impacts on the hydrology of the upperMara River basin, Kenya. Vogel et al. (1999) developed a regionalregression model to analyze the precipitation and temperatureelasticity of the runoff in the United States. Schilling et al. (2010)employed the Generalized Additive Model (GAM) to assess theimpact of land use and land cover change on the discharge in theUpper Mississippi River. Xu et al. (2013) assessed the impacts ofclimate variability and human activities on annual runoff in theLuan River basin, China using the Variable Infiltration Capacity(VIC) model. Bourgault et al. (2014) used the physically-based dis-tributed Mike SHE model to simulate aquifer–peatland–river inter-actions under climate change in southern Quebec, Canada. Moiwoand Tao (2014) analyzed the effects of land use change on the

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 327

hydrological processes of groundwater recharge and discharge in asemiarid area in northeast China using an integrated recharge–dis-charge model driven by WetSpass, MODFLOW, and a Drain Packagein GIS environment. Bulygina et al. (2014) compared four methodsfor estimating flood flows of 5- and 10-year return periods, andflow peaks under both recent land management conditions andspeculative scenarios using the Pontbren catchment, UK as a casestudy and found that the estimated effects vary significantlybetween methods. McIntyre et al. (in press) discussed the chal-lenges of modelling the hydrological impacts of rural land usechange, significant areas of progress and modelling innovations,and proposed priorities for further research.

In addition to these approaches mentioned above, recently theapproach based on the Budyko-type equations, which considerboth the water and energy constraints in hydrological processesover a long-term period in a succinct way, has been widely appliedto quantify or separate the impacts of climate change and humanactivities on runoff (Gardner, 2009; McMahon et al., 2011; Wangand Hejazi, 2011; Roderick and Farquhar, 2011; Teng et al., 2012;Zhan et al., 2013; Xu et al., 2014).

In applying the Budyko-type equations to assess the impacts ofclimate change and human activities on runoff, so far there havebeen two methods developed. The first one is based on the runoffsensitivity concept, in which the sensitivity coefficients of runoffto precipitation and potential evaporation can be calculated sepa-rately via the corresponding partial derivatives (Koster and Suarez,1999; Milly and Dunne, 2002; Li et al., 2007; Ma et al., 2008). Thechange in runoff caused by climate change is estimated by summingthe changes due to both precipitation and potential evaporation,while the contribution of human activities to the change of runoffis regarded as the difference between the observed change in runoffand the estimated change in runoff attributed to climate change.Later, Roderick and Farquhar (2011) built a framework to quantifythe sensitivity coefficients of not only precipitation and potentialevaporation to runoff, but also the parameter in the Budyko-typeequations to runoff, since some Budyko-type equations such asTurc–Pike (Turc, 1954; Pike, 1964; Milly and Dunne, 2002; Yanget al., 2008), Fu (Fu, 1981; Zhang et al., 2004), Zhang (Zhang et al.,2001) and Wang–Tang (Wang and Tang, 2014), provide a parameterto reflect the watershed characteristics and control the shape of theBudyko-type curve. The second approach to apply the Budyko-typeequations to assess the impacts of climate change and human activ-ities on runoff is the decomposition method proposed by Wang andHejazi (2011). The difference between the decomposition methodand the sensitivity method is that the former can independentlyestimate the contributions of either climate change or human activ-ities to runoff change without calculating any sensitivitycoefficients.

There are two issues of concern for the adoption of the Budyko-type equations to assess the impacts of climate change and humanactivities on runoff. The first issue is to determine the form of tem-poral variation in runoff. In many current studies, the form of thevariation in runoff is usually taken as abrupt change by dividingthe whole study period into two or more time periods of multipleyears (Wang and Hejazi, 2011; Roderick and Farquhar, 2011; Wangand Alimohammadi, 2012; Patterson et al., 2013; Xu et al., 2014).The underlying assumption is that either climate change or humanactivities should also take place in the form of abrupt changes, andthe times of these abrupt changes will divide the whole time seriesinto two or more time periods. During each of these time periods,the climatic conditions and intensity of human activities areregarded as relatively stationary, and so are the watershed charac-teristics, which are represented by the parameters in the Budyko-type equations. However, this underlying assumption might fail tocapture the real process of runoff variation, since evolutions of

climate change and human activities would probably be gradualrather than abrupt.

The second issue in the application of the Budyko-type equa-tions is to find the suitable Budyko-type curve to represent thelocal hydro-meteorological conditions and watershed characteris-tics for the study period, which normally means the estimationof the parameters used in the Budyko-type equations. There havebeen some studies to establish the relationship of the long-termaverage values of parameters in the Budyko-type equations ofwatersheds to the factors such as climate, geomorphology, vegeta-tion cover, or proportions of irrigated land and wasteland (Yanget al., 2007; Han et al., 2011; Li et al., 2013). These related factorsare usually treated as constant, but, in fact, some of them such asirrigated land area in the catchment can be changed over years.As a result, the shape of the Budyko-type curve for the catchmentis probably altered.

The goal of this research is to investigate how the parameters inthe Budyko-type equations, would evolve with climate change andhuman activities by establishing the relationships of parameters tothe factors of climate and human activities by using an 11-yearmoving window. With the established relationships, both thedecomposition method (Wang and Hejazi, 2011) and the sensitiv-ity method (Roderick and Farquhar, 2011) are employed to sepa-rate the impacts of climate change and human activities onrunoff. The developed framework is applied to the Weihe River,which is the largest tributary of the Yellow River in China andhas been significantly influenced by both climate change andhuman activities.

The rest of this paper is organized as follows. In the next section,the methods used in the paper are described. The study region anddata sets are described in Section 3. The results and discussion arepresented in Section 4. Finally, the conclusions of this paper aresummarized in Section 5.

2. Methods

The methods to assess the impacts of climate change andhuman activities on runoff are presented below. First, the relation-ships of the parameters in the Budyko-type equations to the factorsof climate and human activities are established based a movingwindow of 11 years. Then, based on the established relationships,the decomposition method and the sensitivity method are appliedto quantify the contributions of climate change and human activi-ties to the change of mean annual runoff in each time window.

2.1. Budyko framework

The water balance equation for a watershed can be written asfollows

P ¼ Q þ Eþ DS=Dt ð1Þ

where P is precipitation, Q is runoff, E is actual evaporation, Dt is thetime step, and DS is water storage change in the watershed. In thelong term, the water storage change can be negligible, i.e. DS = 0.Budyko (1974) demonstrated that in a watershed over a long-termtime scale, the evaporation ratio e, i.e. e = E/P, can be expressed asfollows

e ¼ Bð/Þ ¼ /½1� expð�/Þ� tanhð/�1Þ� �0:5 ð2Þ

where B(�) stands for the Budyko equation; / is the ratio of potentialevaporation to precipitation, i.e. Ep/P, and also called climatic dry-ness index. According to Budyko (1974), two factors limit the evap-oration of the watershed, that if / < 1 the evaporation is limited byenergy supply, and if / > 1 the evaporation is limited by water

Table 1Four single-parameter Budyko-type equations considered in this study.

Budyko-type equation Expression of Budyko-type equation

Turc–Pike (Turc, 1954; Pike, 1964;Milly and Dunne, 2002; Yanget al., 2008)

BTPð/jwTPÞ ¼ ð1þ /�wTP Þ�1=wTP

Fu (Fu, 1981; Zhang et al., 2004)BF ð/jwF Þ ¼ 1þ /� ð1þ /wF Þ

1=wF

Zhang (Zhang et al., 2001) BZ(/|wZ) = (1 + wZ/)(1 + wZ/ + /�1)�1

Wang–Tang (Wang and Tang, 2014) BWT ð/jwWT Þ ¼ ½1þ/�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ/Þ2�4wWT ð2�wWT Þ/p

�½2wWT ð2�wWT Þ�

328 C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338

supply. Besides Eq. (2) proposed by Budyko (1974), there are at leastfour Budyko-type equations, i.e. Turc–Pike (Turc, 1954; Pike, 1964;Milly and Dunne, 2002; Yang et al., 2008), Fu (Fu, 1981; Zhang et al.,2004), Zhang (Zhang et al., 2001), and Wang–Tang (Wang and Tang,2014), each of which provides a parameter w to reflect thewatershed characteristics, and their general expression is denotedby e = B(/|w). Compared with the non-parametric Budyko equation,the single-parameter Budyko-type equations are more flexible toadapt to different watershed characteristics (Xiong and Guo,2012). The details of the four Budyko-type equations consideredin this paper are presented in Table 1, where the parameters inthe Budyko-type equations of Turc–Pike, Fu, Zhang, and Wang–Tangare denoted as wTP, wF, wZ and wWT, respectively.

2.2. Estimating w by the covariate analysis

Moving window is a simple but effective way to smooth ran-dom variations in hydrological series and has been widely usedto present the nonstationarities in the series (Gilroy and McCuen,2012; Willems, 2013). In this paper, in order to remove the influ-ences of both water storage change and natural climate variations,an 11-year width of the time window is applied to the resultingtime series of mean annual P, Ep, E and Q. Choosing a suitable win-dow width is a critical issue, but there is not a specific guideline. Ingeneral, if the annual series is long, a wider time window should betaken. In this study, considering the length of the hydrological ser-ies and the denotation of the hydrological variables, the time win-dow of 11 years is taken. The mean annual precipitation, potentialevaporation, actual evaporation and runoff in the time windowcentered with year t are denoted by Pt, Ept, Et and Qt, respectively.

In order to reveal how the parameter w in each Budyko-typeequation would vary with climate change and human activities,w is expressed as the dependent variable of some explanatory vari-ables via a general linear function, as follows

wt ¼ b0 þ b1xc1;tþ; � � � ;þbkxc

k;t þ bkþ1xhkþ1;tþ; � � � ;þbmxh

m;t ð3Þ

where wt is the parameter in each Budyko-type equation of year t,xc

1;t ; . . . ; xck;t; x

hkþ1;t; . . . ; xh

m;t are m explanatory variables in the timewindow centered with year t and denoted by vector Xt, amongwhich xc

i;t (i = 1, 2, . . . , k) is the explanatory variable about climateconditions, such as mean annual precipitation, temperature andpotential evaporation in the time window; and xh

i;t (i = k + 1, . . . , m)is the explanatory variable about human activities. b0, b1, . . . , bm

are m + 1 regression parameters, denoted by vector b. It should benoted that, as Eq. (3) is a general linear function, the terms of xc

i;t

or xhi;t can not only take the identity function form of the variables

of climate factors or human activities, but also take other functionforms such as logarithmic and exponential forms in order todescribe the nonlinear relationships of the parameter w to climatefactors or human activities. If no explanatory variables are intro-duced into Eq. (3), then it means that the parameter w is treatedas a constant over the whole period.

Substituting Eq. (3) in the Budyko-type equations shown inTable 1, the observed mean annual runoff Qt in each time window

can be expressed as a function of the variables of climatic factorsand human activities, as follows

Qt ¼ Pt ½1� Bð/tjwtÞ� þ et ¼ Pt ½1� Bð/t;Xt jbÞ� þ et ð4Þ

where /t is the climatic dryness index of the time window centeredwith year t, and et is the model residual. The residual et is assumedto follow a normal distribution et � N(0, r2), whose mean and stan-dard deviation are 0 and r, respectively, i.e., Qt � N(Pt[1 � B(/t, Xt|-b)], r2). The parameters b and r are estimated by using maximumlikelihood estimation method (MLE) with the likelihood functionwith respect to b and r given by

Lðb;rÞ ¼Yn

t¼1

f ðQ tjPt ½1� Bð/t;Xt jbÞ�;rÞ ð5Þ

where f(�) stands for the density function of a normal distribution,and n is the length of the observed mean annual runoff time seriesover the moving windows. By maximizing the likelihood functionexpressed as Eq. (5), the estimated parameter vector b and standarddeviation r are obtained. Thus, the estimated annual parameter wt

is given by wt ¼ bXTt , and the simulated mean annual runoff of each

time window is obtained as follows

bQ t ¼ Pt ½1� Bð/t;Xt jbÞ� ð6Þ

In practice, we need to select proper covariates of w from anumber of candidate explanatory variables of climate or humanactivities. In this paper, the selection of the covariates of w is per-formed by using the stepwise regression (Burnham and Anderson,2002) with Akaike’s information criterion (AIC; Akaike, 1974) asthe model selection criterion, which is given as follows

AIC ¼ �2 ln Lðb; rÞ þ 2ðmþ 2Þ ð7Þ

where m + 2 is the total number of parameters used in Eq. (5).

2.3. Separating the impacts of climate change and human activities onrunoff

The Budyko-type equation with the parameter w expressed inthe form of Eq. (3) is employed as the physical basis to separatethe impacts of climate change and human activities on runoff. Inthe pre-change period, the precipitation, potential evaporationand runoff are denoted by P0, Ep0 and Q0, following the Budyko-type equation with the parameter w0, whose explanatory variablevector is ðxc

1;0; . . . ; xck;0; x

hkþ1;0; . . . ; xh

m;0Þ. In the post-change period, theprecipitation, potential evaporation and runoff are P1 = P0 + DP,Ep1 = Ep0 + DEp and Q1 = Q0 + DQ, following the Budyko-type equa-tion with the parameter w1 = w0 + Dw, whose explanatory variablevector is ðxc

1;1; . . . ; xck;1; x

hkþ1;1; . . . ; xh

m;1Þ, being equal toðxc

1;0 þ Dxc1; . . . ; xc

k;0 þ Dxck; x

hkþ1;0 þ Dxh

kþ1; . . . ; xhm;0 þ Dxh

mÞ. The twomethods to separate the contributions of climate change andhuman activities (denoted by DQc and DQh respectively) to therunoff change DQ are described as follows.

2.3.1. Decomposition methodAccording to the Budyko-type equations displayed in Table 1,

the evaporation ratio E/P is determined by both the climatic dry-ness index Ep/P and the parameter w. Fig. 1 describes all eight pos-sible shift directions of the relationship between E/P and Ep/P fromthe current point O. It can be seen that the vertical shifts to direc-tion 1 or 5 are induced only by the change of w, while the shiftsfrom O along the Budyko-type curve to direction 3 or 7 are inducedonly by the change of Ep/P. However, the shifts to other four direc-tions are induced by the simultaneous changes of w and Ep/P, andhow to separate the impacts on runoff of the simultaneous changesof w and Ep/P has become a very critical problem. To address this

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

E/ P

Ep /P

Water limitWater limitEp/P ↑, w↑

w↑

Ep/P ↑Ep/P ↓, w↑

Ep/P ↓

w↓

Ep/P ↓, w↓

Ep/P ↑, w↓

O

12

3

45

6

7

8

2w =

1.5w =

2.5w =

Fig. 1. Eight possible shift directions of the relationship between evaporation ratioand climatic dryness index (Fu equation).

Ep/P

E/P

Ener

gy li

mit

Fig. 2. Decomposition method to separate the contributions of the changes inclimatic dryness index and parameter w to the change of evaporation ratio.

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 329

issue, Wang and Hejazi (2011) proposed the decompositionmethod.

The decomposition method is presented in Fig. 2. In the pre-change period, the relationship between evaporation ratio and cli-matic dryness index is represented by point A (Ep0/P0, E0/P0), whichis located on the Budyko-type curve of w0. In the post-change per-iod, the relationship between evaporation ratio and climatic dry-ness index has shifted to point B (Ep1/P1, E1/P1), which is locatedon the Budyko-type curve of w1. Unlike the original descriptionof the decomposition method (Wang and Hejazi, 2011) where itonly showed the Budyko-type curve of the pre-change period, inthis case the Budyko-type curve of the post-change period, i.e.the curve of w1, is also displayed in the figure. Point C (Ep1/P1,E01=P1) is a hypothetical point, which is located on the same Bud-yko-type curve as point A but has the same climatic dryness indexwith point B. The shift from point A to point B is assumed to firstevolve from point A to point C along the Budyko-type curve ofw0, and then move vertically from point C to point B. Thus, thechange in the evaporation ratio of the post-change period (pointB) relative to that of the pre-change period (point A), i.e. DeA?B =E1/P1 � E0/P0, can be separated into two parts, i.e. the differenceof evaporation ratio between points A and C denoted byDeA!C ¼ E01=P1 � E0=P0, and the difference of evaporation ratiobetween points C and B denoted by DeC!B ¼ E1=P1 � E01=P1, whichare attributed to the changes of climatic dryness index and w,respectively. Accordingly, the difference of the runoff between

point A and point B, i.e. DQ, can be divided into two parts via pointC, denoted by DQA?C and DQC?B, which are attributed to thechange of hydrological inputs (precipitation and potential evapora-tion) and the change of parameter w, respectively, and computedby

DQA!C ¼ P1½1� Bð/1;w0Þ� � P0½1� Bð/0;w0Þ� ð8Þ

DQC!B ¼ P1½Bð/1;w0Þ � Bð/1;w1Þ� ð9Þ

where /0 = Ep0/P0 is the climatic dryness index of the pre-changeperiod, and /1 = Ep1/P1 is the climatic dryness index of the post-change period.

In previous studies (Wang and Hejazi, 2011; Patterson et al.,2013), DQA?C and DQC?B were treated as the contributions of cli-mate change and human activities to DQ, respectively, under theassumption that the change of hydrological inputs is attributedto climate change while the change of w is attributed to humanactivities. However, it might not be reasonable to attribute thechange of the parameter w just to human activities, since climaticconditions could also be an important factor affecting the parame-ter w (Yang et al., 2007; Williams et al., 2012). Therefore, the runoffchange induced by the change of w, i.e. DQC?B, should be attrib-uted not only to human activities but also to climate change.According to Eq. (3), the difference between w1 and w0 can beexpressed as

Dw ¼ w1 �w0 ¼Xk

i¼1

biDxci þ

Xm

i¼kþ1

biDxhi ¼ Dwc þ Dwh ð10Þ

where Dwc ¼Pk

i¼1biDxci , and Dwh ¼

Pmi¼kþ1biDxh

i are the contribu-tions of climate change and human activities to Dw. The termDQC?B, caused by the change of w, thus can be divided into twoparts, according to the ratios of Dwc and Dwh to Dw, i.e. the humanactivities-induced part is (Dwh/Dw)DQC?B and the climate change-induced part (Dwc/Dw)DQC?B.

After separating DQC?B into two parts, now the total contribu-tion to the change of runoff induced by climate change, denotedby DQc, should contain two components, i.e. DQc_1 and DQc_2.DQc_1 is DQA?C induced by the change of the hydrological inputs(precipitation and potential evaporation) to the watershed, whileDQc_2 is the climate change-induced part of DQC?B. Mathemati-cally, these terms are expressed as:

DQc ¼ DQ c 1 þ DQ c 2 ð11Þ

DQc 1 ¼ P1½1� Bð/1;w0Þ� � P0½1� Bð/0;w0Þ� ð12Þ

DQc 2 ¼ Dwc

DwP1½Bð/1;w0Þ � Bð/1;w1Þ� ð13Þ

For DQc_2, the contribution due to the variation of climatic explan-atory variable xc

i (i = 1, . . . , k) can be estimated as

DQc 2 xci ¼ ðbiDxc

i =DwÞDQ C!B ð14Þ

The contribution to the change of runoff induced by humanactivities is the part of DQC?B, and is computed as:

DQh ¼ Dwh

DwP1½Bð/1;w0Þ � Bð/1;w1Þ� ð15Þ

For DQh, the contribution due to the variation of human explanatoryvariable xh

i (i = k + 1, . . . , m) can be estimated as

DQh xhi ¼ ðbiDxh

i =DwÞDQC!B ð16Þ

Finally, the estimated runoff change for the decompositionmethod, denoted by DQDM, is the sum of DQc in Eq. (11) and DQh

in Eq. (15).

Fig. 3. The map of the Weihe River basin (the catchment above the Huaxian gauge).

330 C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338

2.3.2. Sensitivity methodAccording to Roderick and Farquhar (2011) and Donohue et al.

(2011), the total differential of runoff Q can be expressed as

dQ ¼ @Q@P

dP þ @Q@Ep

dEpþ @Q@w

dw ð17Þ

where the partial derivatives in each term on the right hand side ofequation are the so-called sensitivity coefficients of runoff to pre-cipitation, potential evaporation and the parameter w, respectively.With the definitions of the conditions of climate and human activ-ities in the pre-change period and the post-change period above, thediscretized forms of Eq. (17) can be expressed as

DQ ¼ @Q 0

@P0DP þ @Q0

@Ep0DEpþ @Q 0

@w0Dw ð18Þ

In Eq. (18), the gradients of the intervals from (P0, Ep0, w0) to(P1, Ep1, w1) are represented by three sensitivity coefficients at point(P0, Ep0, w0), i.e. @Q0/@P0, @Q0/@Ep0 and @Q0/@w0. Thus, Eq. (18)strictly holds only for (DP, DEp, Dw) ? (0, 0, 0). If the absolute val-ues of DP, DEp or Dw are too large, the estimated value of DQ by Eq.(18) may have a significant deviation from the observed DQ. Inorder to reduce the discretization error, the means of the sensitivitycoefficients in the pre-change and post-change periods, i.e. (@Q0/@P0 + @Q1/@P1)/2, (@Q0/@Ep0 + @Q1/@Ep1)/2 and (@Q0/@w0 + @Q1/@w1)/2, are taken to approximate the gradients of the interval from(P0, Ep0, w0) to (P1, Ep1, w1).

Considering that Dw = Dwc + Dwh as shown by Eq. (10), the con-tribution to the change of runoff induced by climate change DQc isfinally calculated as follows

DQ c ¼ DQ c 1 þ DQc 2 ð19Þ

DQ c 1 ¼ 12

@Q 0

@P0þ @Q1

@P1

� �DP þ 1

2@Q 0

@Ep0þ @Q 1

@Ep1

� �DEp ð20Þ

DQ c 2 ¼ 12

@Q 0

@w0þ @Q 1

@w1

� �Dwc ð21Þ

Similar to Eqs. (12) and (13), DQc_1 is attributed to the changes ofthe hydrological inputs to the watershed, and DQc_2 is attributedto the change of w induced by climate change. The contributiondue to the variation of climatic explanatory variable xc

i (i = 1, . . . , k)to DQc_2 can be estimated as

DQ c 2 xci ¼ biDxc

i ð@Q 0=@w0 þ @Q 1=@w1Þ=2 ð22Þ

The contributions to the change of runoff induced by humanactivities is calculated as follows

DQ h ¼ 12

@Q 0

@w0þ @Q1

@w1

� �Dwh ð23Þ

The contribution due to the variation of human explanatoryvariable xh

i (i = k + 1, . . . , m) to DQh can be estimated as

DQ h xhi ¼ biDxh

i ð@Q 0=@w0 þ @Q 1=@w1Þ=2 ð24Þ

For the sensitivity method, the estimated runoff change,denoted by DQSM, is the sum of DQc in Eq. (19) and DQh in Eq. (23).

3. Study area and data

3.1. The Weihe River basin

As the largest tributary of the Yellow River, the Weihe River(Fig. 3) flows through the southern Loess Plateau with a drainagearea of 134,800 km2, which is the transitional region from asemi-humid region to an arid region. With the typical continentalmonsoon climate, the precipitation in the Weihe River basin has a

strong seasonality with more than 60% of annual precipitation con-centrated in the flood season between July and October. Influencedby global climate change, both annual precipitation and tempera-ture of this basin have been found to present obvious trends duringthe past decades (Du and Shi, 2012).

As an important economic and agricultural zone, the WeiheRiver has been drastically affected by human activities, especiallyin recent decades. The influence of human activities on the WeiheRiver basin can be mainly attributed to urbanization, irrigated agri-culture, and water and soil conversion projects. These activitieshave significant influences on the hydrological processes in theWeihe River basin (Zhao et al., 2013; Zuo et al., 2014).

3.2. Data

In this study, the annual runoff data from 1960 to 2009 areobtained from the Huaxian gauge on the Weihe River. The Huaxiangauge is located 73 km upstream from the outlet of the WeiheRiver. The drainage area at the station is 106,500 km2 and accountsfor about 80% of the total drainage area of the Weihe River basin.The annual precipitation and annual average temperature areobtained from the daily data of the 21 meteorological stations inor around the study area from 1960 to 2009. The annual potentialevaporation of 1960–2009 is estimated by the modified Hargreavesequation (Droogers and Allen, 2002; Adam et al., 2006) using themonthly values of precipitation, temperature and daily tempera-ture range. In this study, the mean annual runoff, precipitationand potential evaporation in each of the 11-year moving windowsare calculated from the annual data. The actual evaporation in eachtime window is estimated as the difference between precipitationand runoff assuming that the water storage of the basin is negligi-ble, i.e. DS = 0 (Yang et al., 2007; Roderick and Farquhar, 2011;Patterson et al., 2013). To reveal the impact of climate change onthe parameter w in each Budyko-type equation, the mean annualprecipitation, temperature and potential evaporation are used asthe candidates of climatic explanatory variables.

In order to quantify the impact of human activities on theannual runoff of the Weihe River, four indices, i.e. population, grossdomestic product, irrigated area, and cultivated land area are con-sidered. These factors are all important indices for the alterationsof land use and land cover of catchment by human activities espe-cially in the fast developing countries like China, and have beenemployed to assess the impact of human activities on runoff(Wang and Hejazi, 2011; Chen et al., 2014). Corresponding to thetime scale of hydrological data, the means of the population, grossdomestic product, cultivated land area, and irrigated area in eachof 11-year moving windows are used as the candidates of humanexplanatory variables for describing w, and denoted by Pop, GDP,

11-year moving average

19

24

29

34

39

1960 1970 1980 1990 2000 2010Year

1

5

25

125

625

1960 1970 1980 1990 2000 2010Year

2.5

3

3.5

4

4.5

1960 1970 1980 1990 2000 2010Year

0.2

0.4

0.6

0.8

1

1.2

1960 1970 1980 1990 2000 2010Year

Popu

latio

n (1

06 )C

ultiv

ated

land

are

a (1

06 hm

2 )

Irrig

ated

are

a (1

06 hm

2 )9

Gro

ss d

omes

tic

(pr

oduc

t10

)Annual value

(a) (b)

(c) (d)

Fig. 4. Evolutions of four variables representing human activities. (a), (b), (c) and (d) are for population, gross domestic production, cultivated land area and irrigated area ofthe Weihe River basin, respectively.

Table 2Results of trend analysis of the hydro-meteorological series. The symbol ‘⁄’ indicatesthat the series can pass the Mann–Kendall trend test at the 0.05 significance level.

Series ZMK Trend

Runoff �3.68⁄ ;Precipitation �1.94 ;Temperature 4.12⁄ "Potential evaporation 0.42 "

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 331

CA, and IA, respectively. Since the majority of the cities, populationand agricultural regions of the Weihe River basin are distributed inthe Shaanxi Province, Pop, GDP, CA, and IA for the Weihe River basinare represented by data from this province (Yang, 2009). FromFig. 4, it can be found that Pop and GDP present an obvious increas-ing trend, while CA presents an obvious decreasing trend. IAexpands rapidly in the 1960s and 1970s, but after the 1980s, it pre-sents a slight decreasing trend.

4. Results and discussion

4.1. Trend analysis of the annual hydro-meteorological series

To detect the long-term variation of runoff together with cli-mate change of the Weihe River basin, the nonparametric Mann–Kendall method (Mann, 1945; Kendall, 1975) is used to examinethe trends in the annual runoff, precipitation, average temperature,and potential evaporation series of 1960–2009. As shown inTable 2, the annual runoff presents a decreasing trend at the 0.05significance level, while the annual average temperature presentsan increasing trend at the 0.05 significance level. The annual pre-cipitation also presents some degree of decreasing trend, but can-not pass the Mann–Kendall trend test at the 0.05 significance level.The annual potential evaporation has no significant trend. Fig. 5also shows the linear trends and 11-year moving averages of theseseries.

4.2. Estimation of the annual values of w in the Budyko-type equations

The annual value of the parameter w from 1965 to 2004,denoted by wannual, in each of the four Budyko-type equations,i.e., Turc–Pike, Fu, Zhang, and Wang–Tang, is computed with theobserved mean annual evaporation, precipitation and potentialevaporation in each time window as inputs. Fig. 6 displays the evo-lution and linear trend of wannual in each Budyko-type equationcompared with the estimated value of w treated as a constant forall time windows. From this figure, the value of wannual in the period

of the 1960s and early 1970s is larger than the constant estimationof w, while in the periods 1990s and 2000s, wannual is smaller thanthe constant estimation of w. It also can be found that all theannual parameters in the four Budyko-type equations presentobvious increasing trends, which would make the Budyko-typecurves for the Weihe River basin moving up. As the result, forthe same hydrological inputs of precipitation and potential evapo-ration, the evaporation ratio will increase and the runoff willdecrease. So the decrease of the annual runoff of the Weihe Riveris due to not only the change of hydrological inputs, but also thechange of the relation of water-energy of the basin.

4.3. Relationships of w with explanatory variables

The covariates analysis is performed to build the relationshipbetween the parameter w in each Budyko-type equation andexplanatory variables. In order to quantify the relationshipbetween w and explanatory variables, three functional forms (i.e.,identity, exponential, and logarithmic) are considered for each ofthe explanatory variables. The mean values of the precipitation,temperature, potential evaporation, population, gross domesticproduction, cultivated land area and irrigated area of the WeiheRiver basin over the whole observed period are calculated anddenoted by P, T , Ep, Pop, GDP, CA and IA, respectively. All these can-didate explanatory variables are normalized by the mean values inorder to eliminate the impacts of different physical dimensions onthe results. Using the stepwise regression method with AIC as the

Linear trend of annual hydrological time seriesAnnual hydrological time series

y = -1.1146x + 2272.623p = 0.000

020406080

100120140160180200

1960 1970 1980 1990 2000 2010

Year

y = -1.864x + 4234.127p= 0.045

300350400450500550600650700750800

1960 1970 1980 1990 2000 2010

Year

y = 0.026x - 41.586p = 0.000

8

8.5

9

9.5

10

10.5

11

1960 1970 1980 1990 2000 2010

Year

y = 0.533x - 152.168p = 0.406

700

750

800

850

900

950

1000

1050

1100

1960 1970 1980 1990 2000 2010

Year

Run

off (

mm

)Te

mpe

ratu

re (°

C)

Prec

ipita

tion

(mm

)Po

tent

ial e

vapo

ratio

n (m

m)

(a) (b)

(c) (d)

11- year moving average of annual hydrological time series

Fig. 5. The hydrological time series of the Weihe River. (a), (b), (c) and (d) are the runoff, precipitation, temperature and potential evaporation of the Weihe River,respectively. p value is the significance level of the slope of the linear regression.

y = 0.011x - 18.22p= 0.000

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

1965 1975 1985 1995 2005

y = 0.010x - 17.932p = 0.000

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

1965 1975 1985 1995 2005

y = 0.033x - 63.689p = 0.000

1

1.5

2

2.5

3

3.5

1965 1975 1985 1995 2005

y = 0.002x - 4.037p = 0.000

0.55

0.6

0.65

0.7

0.75

1965 1975 1985 1995 2005

in T

urc-

Pike

equ

atio

nw

Year

in F

u eq

uatio

nw

Year

in Z

hang

equ

atio

nw

Year

in W

ang-

Tang

equ

atio

nw

Year

(a) (b)

(c) (d)

ˆ annualw ˆLinear trend of annualwConstant estimation of w

Fig. 6. Evolution and linear trend of the annual parameter w in each of the four Budyko-type equations, i.e., (a) Turc–Pike, (b) Fu, (c) Zhang, and (d) Wang–Tang. p value is thesignificance level of the slope of the linear regression.

332 C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338

selection criterion, the covariate variables for w are determined,and shown in Table 3. The estimated parameter w for each of theBudyko-type equations of Turc–Pike, Fu, Zhang, and Wang–Tangare respectively given as follows:

wTPt ¼ �0:320þ 0:688 expðTt=TÞ þ 0:270 expðEpt=EpÞ

þ 0:247 lnðIAt=IAÞ ð25aÞ

wFt ¼ �0:172þ 0:720 expðTt=TÞ þ 0:322 expðEpt=EpÞ

þ 0:255 lnðIAt=IAÞ ð25bÞ

wZt ¼ �7:428þ 2:473 expðTt=TÞ þ 1:109 expðEpt=EpÞ

þ 0:596 lnðIAt=IAÞ ð25cÞ

Table 3Covariate analysis for the parameter w in each Budyko-type equation. RMSE, RE and NSE stand for root-mean-square error, relative error and Nash–Sutcliffe efficiency coefficient,respectively. The symbol ‘–’ indicates that no covariates are introduced and w is considered as constant.

Budyko-type equation Covariates AIC RMSE RE (%) NSE

Turc–Pike – 265.95 6.39 0.7 0.84

expðTt=TÞ; expðEpt=EpÞ; lnðIAt=IAÞ 170.34 1.80 �0.0 0.99

Fu – 268.42 6.60 0.8 0.83

expðTt=TÞ; expðEpt=EpÞ; lnðIAt=IAÞ 169.06 1.77 �0.0 0.99

Zhang – 275.60 7.21 1.3 0.80

expðTt=TÞ; expðEpt=EpÞ; lnðIAt=IAÞ 168.28 1.75 �0.0 0.99

Wang–Tang – 278.52 7.48 1.2 0.78

lnðTt=TÞ; lnðEpt=EpÞ; lnðIAt=IAÞ 165.99 1.70 �0.0 0.99

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 333

wWTt ¼ 0:669þ 0:401 lnðTt=TÞ þ 0:304 lnðEpt=EpÞ þ 0:061

� lnðIAt=IAÞ ð25dÞ

In Eq. (25), the parameter w in each Budyko-type equation is relatedto the factors of both climate conditions and human activities. Forall Budyko-type equations, temperature, potential evaporation andirrigated area are selected to be the covariates of w. The findingon the positive relationship between w and irrigated area is consis-tent with the results of Han et al. (2011). As shown in Fig. 7, the esti-mated parameter w for each Budyko-type equation by Eq. (25) isconsistent with the corresponding value of wannual.

Besides the model selection criterion of AIC, root-mean-squareerror (RMSE), relative error (RE) of the volumetric fit betweenthe observed mean annual runoff and the simulated, and Nash–Sutcliffe efficiency coefficient (NSE; Nash and Sutcliffe, 1970) arealso used to assess the performance of the four Budyko-type equa-tions in simulating the mean annual runoff. From Table 3 andFig. 8, it can be seen that the performance of all Budyko-type equa-tions in simulating the mean annual runoff has been obviouslyimproved due to the introduction of the covariates for w. Specifi-cally, with no covariate introduced, i.e. the parameter w treated

y = 0.918x + 0.185R² = 0.930

2

2.1

2.2

2.3

2.4

2.5

2.6

2 2.1 2.2 2.3 2.4 2.5 2.6

y = 0.927x + 0.163R² = 0.950

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

,ˆ annual TPw

ˆTP w

,ˆ annual Zw

ˆZw

(a)

(c)

Fig. 7. Comparison between the estimated parameter w by Eq. (25) and the annual valueand Wang–Tang, respectively.

as a constant, the NSE of each Budyko-type equation varies from0.78 to 0.84, while with the explanatory variables introduced asthe covariates of w, the NSEs for all Budyko-type equations reachup to 0.99. In addition, according to the values of AIC or RMSE,the Wang–Tang equation has the best quality of fitting effect, fol-lowed by the equations of Zhang, Fu and Turc–Pike, respectively.

As shown in Fig. 8, with the parameter treated as a constant,each of the four Budyko-type equations has a poor performancein simulating the mean annual runoff in both early phase and latephase of the observed period. Particularly, in the period of 1960sand early 1970s, the simulated mean annual runoffs are smallerthan those observed, while in the period of 1990s and 2000s, theopposite is the case. This finding indicates that a constant param-eter in the Budyko-type equation is not able to capture the catch-ment properties in different periods.

4.4. Assessing the impacts of climate change and human activities onrunoff

Based on the relationship between the parameter w in eachBudyko-type equation and the explanatory variables, i.e. Eq. (25),the decomposition method and the sensitivity method are

y = 0.925x + 0.223R² = 0.937

2.7

2.8

2.9

3

3.1

3.2

3.3

2.7 2.8 2.9 3 3.1 3.2 3.3

y = 0.944x + 0.037R² = 0.952

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74

(b)

,ˆ annual Fw

ˆFw

,ˆ annual WTw

ˆWT

w

(d)

wannual . (a), (b), (c) and (d) are for the Budyko-type equations of Turc–Pike, Fu, Zhang,

30

40

50

60

70

80

90

100

1965 1970 1975 1980 1985 1990 1995 2000 2005

Mea

n an

nual

runo

ff (m

m)

Year

30

40

50

60

70

80

90

100

1965 1970 1975 1980 1985 1990 1995 2000 2005

Mea

n an

nual

runo

ff (m

m)

Year

30

40

50

60

70

80

90

100

1965 1970 1975 1980 1985 1990 1995 2000 2005

Mea

n an

nual

runo

ff (m

m)

Year

30

40

50

60

70

80

90

100

1965 1970 1975 1980 1985 1990 1995 2000 2005

Mea

n an

nual

runo

ff (m

m)

Year

0.84NSE =

0.99NSE =

0.83NSE =

0.99NSE =

0.80NSE =

0.99NSE =

0.78NSE =

0.99NSE =

Simulated mean annual runoff with w estimated by Eq. (25)

Observed mean annual runoff Simulated mean annual runoff with w treated as a constant

(a) (b)

(c) (d)

Fig. 8. Comparison between the simulated mean annual runoff and the observed mean annual runoff from the cases when: the parameter w in each Budyko-type equation istreated as a constant, and the value of parameter w in each Budyko-type equation is estimated by Eq. (25). (a), (b), (c) and (d) are for the Budyko-type equations of Turc–Pike,Fu, Zhang, and Wang–Tang, respectively.

334 C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338

employed to quantify the change of mean annual runoff in eachtime window relative to the mean annual runoff of the first timewindow 1960–1970 centered at 1965. In other words, the periodof 1960–1970 is treated as the pre-change period and the bench-mark, while each of the other time windows is individually treatedas the post-change period.

Fig. 9 shows the evolutions of climate-induced runoff changeDQc and human-induced runoff change DQh in the period of1965–2004 (DQc and DQh for 1965, i.e. the time window of1960–1970, are both set to 0) estimated by the decompositionmethod and the sensitivity method on the basis of the covariateanalysis for the parameter w in each Budyko-type equation. Forall Budyko-type equations, the two methods yield close results thatclimate change and human activities are both the driving factors tothe decline of runoff of the Weihe River and climate change has amore profound impact than human activities.

In the latter parts of this section, we focus on the results basedon the Wang–Tang equation to further discuss the impact of cli-mate change and human activities on runoff of the Weihe River.Table 4 summaries the DQc, DQh, and DQ estimated by the decom-position method and the sensitivity method based on the Wang–Tang equation together with the observed mean annual runoffand the change of the observed mean annual runoff DQobs in differ-ent time windows. In this case, we select five time windows, i.e.1960–1970, 1970–1980, 1980–1990, 1990–2000, and 1999–2009,to stand for the periods of the 1960s, 1970s, 1980s, 1990s, and2000s, respectively. From this table, the impacts of climate changeor human activities on runoff of the Weihe River vary over the peri-ods. According to the results of the decomposition method, theestimated DQc in the 1970s is �23.8 mm, accounting for 70% ofDQDM. While the estimated DQc in the 1980s increases to�1.3 mm, only accounting for 8% of DQDM. In the periods of the1990s and 2000s, the estimated DQc are �39.7 mm and�40.0 mm, whose contributions to DQDM are both about 80%. Sincethe parameter w is positively related to the irrigated area in thebasin, the temporal variation of the estimated DQh matches that

of irrigated area. In the 1970s, the estimated DQh is �10.0 mm,and then it decreases to �14.4 mm in the 1980s. In the 1990sand 2000s the estimated DQh increases to �10.2 mm and�9.9 mm, respectively. The contributions to DQDM in the four peri-ods are 30%, 92%, 20% and 20%, respectively. Consequently, climatechange should play the dominant role in leading to the decline ofrunoff in the Weihe River during most of the periods. The sameconclusion is drawn from the sensitivity method.

Fig. 10 shows the two components of DQc based on the Wang–Tang equation, i.e. DQc_1 induced by the change of the hydrologicalinputs of both precipitation and potential evaporation to thewatershed, and DQc_2 induced by climate change via altering thewatershed characteristics represented by the parameter w. It canbe found that DQc_1 plays the decisive role in leading to the changeof mean annual runoff, but the impact of DQc_2 cannot be ignored.

It has been known that the parameter w in the Wang–Tang equa-tion is related to both the climatic factors of potential evaporationand temperature. In order to quantify the individual impacts of thevariations in these two factors on runoff of the Weihe River via alter-ing w, DQc_2 is divided into potential evaporation-induced compo-nent denoted by DQc_2_Ep, and temperature-induced componentdenoted by DQc_2_T. Fig. 11 displays DQc_2_Ep and DQc_2_T estimatedby both the decomposition method and the sensitivity method asexpressed by Eqs. (14) and (22). From this figure, DQc_2_Ep presentsa wave-like variation similar with potential evaporation, andDQc_2_T

presents a significant decreasing trend after the late 1980s. Thisfinding reveals that the temperature rise induces a significantdecrease in runoff of the Weihe River through increasing the valueof w. This finding is consistent with some previous studies (Du andShi, 2012; Xiong et al., 2014), where it has been found that theannual runoff of the Weihe River is negatively related to the temper-ature and the temperature rise is an important reason for thedecrease of the runoff. In most previous studies based on the Budykoframework, the effect of temperature on runoff has been seldomtaken into consideration or temperature is assumed to play its rolein changing runoff indirectly via altering potential evaporation

Fig. 9. Contributions of climate change (left column) and human activities (right column) to the variation of runoff estimated by the decomposition method and thesensitivity method. DQc and DQh for each year are estimated benchmarked with the mean annual runoff in the time window of 1960–1970. (a), (b), (c) and (d) are for theBudyko-type equations of Turc–Pike, Fu, Zhang, and Wang–Tang, respectively.

Table 4DQc and DQh in different time windows estimated by the decomposition method and the sensitivity method based on the Wang–Tang equation. DQc and DQh for each timewindow are estimated benchmarked with the mean annual runoff in the time window of 1960–1970.

Time window Decomposition method (mm) Sensitivity method (mm) Observed runoff (mm) DQobs (mm)

DQc DQh DQDM DQc DQh DQSM

1960–1970 0 0 0 0 0 0 89.98 01970–1980 �23.9 �10.0 �33.9 �22.9 �10.9 �33.8 55.4 �34.61980–1990 �1.3 �14.2 �15.5 �1.1 �14.4 �15.5 74.3 �15.71990–2000 �39.7 �10.2 �49.9 �37.8 �11.9 �49.6 40.5 �49.51999–2009 �40.0 �9.9 �50.0 �38.3 �11.3 �49.6 41.7 �48.3

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 335

_1cQ_ 2cQ

_1cQ_ 2cQ

Δ

cQΔ

cQ

Δ

Δ

ΔΔ

(a)

(b)

Fig. 10. Two components of DQc based on the Wang–Tang equation, i.e. DQc_1

induced by the change of hydrological input, and DQc_2 induced by the change ofthe parameter w: (a) is the result of the decomposition method; (b) is the result ofthe sensitivity method. DQc_1 and DQc_2 are estimated benchmarked with the meanannual runoff in the time window of 1960–1970.

Δ

Δ

Δ

ΔΔ

_2

cQ

Δ_

2c

Q

_ 2 _c TQ_ 2 _c EpQ

_ 2 _c TQ_ 2 _c EpQ

(a)

(b)

Fig. 11. Two components of DQc_2 based on the Wang–Tang equation, i.e. potentialevaporation-induced component DQc_2_Ep, and temperature-induced componentDQc_2_T: (a) is the result of the decomposition method; (b) is the result of thesensitivity method. DQc_2_Ep and DQc_2_T are estimated benchmarked with the meanannual runoff in the time window of 1960–1970.

336 C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338

(Gardner, 2009). This finding highlights the importance of consider-ing the effect of the temperature changes on the runoff throughaltering the parameter in the Budyko-type equation.

4.5. Discussion of actual evaporation estimation

One thing that is very important to be discussed is the estima-tion of actual evaporation. It is known that actual evaporation isthe essential information for estimating the parameters w in theBudyko-type equations. However, actual evaporation is perhapsthe most difficult hydrological process to be estimated accurately,and therefore there are many methods available including long-term water balance method (Xu and Singh, 2004), remote sensingmethod (Leuning et al., 2008; El Tahir et al., 2012), Budyko-typemethod (Wang and Tang, 2014), complementary relationshipmethod (Xu and Singh, 2005), and the modified Thornthwaitewater balance method (Gao et al., 2012). Each method has itsown advantages and drawbacks, and ideally it would be better touse the observed actual evaporation data. However, due to the factthat observation data of regional evaporation are usually not avail-able, long-term water balance equation has been used as the onlymethod to validate all other methods mentioned above (Gao et al.,2012). In other words, long-term water balance equation should bethe most reliable method to estimate actual evaporation. Giventhat long-term water storage change of the catchment is negligible,the actual evaporation data used in this study as in many previousstudies (e.g. Yang et al., 2007; Roderick and Farquhar, 2011;Patterson et al., 2013) are estimated by the water balance equationexpressed as E = P � Q.

In this paper, the estimation of actual evaporation by usingE = P � Q is just a way to obtain the data of actual evaporation,and is not a part of the proposed framework for separating theimpacts of climate change and human activities on runoff.Although the actual evaporation had better be estimated indepen-dent of runoff observation data, in practice the actual evaporationestimated as E = P � Q could be inevitably influenced by the obser-vation of runoff. This might lead to a circular logic that runoffobservation data is used to estimate actual evaporation, then theestimated actual evaporation is used to estimate w, and in returnthe estimated w is used to investigate changes in runoff. To avoidthis circular logic, alternative evaporation estimation methods thatare reliable but do not involve runoff observation data should beconsidered in future studies, and the influence of different evapo-ration estimation methods on the estimation of parameter wshould be investigated.

5. Conclusions

In this paper, we present a framework based on four Budyko-type equations with time-varying parameters to assess the impactsof climate change and human activities on runoff by using an 11-year moving window. The relationship between the parameter win each Budyko-type equation and the variables of climate condi-tions and human activities is built to model the temporal variationof w. Based on this relationship, the decomposition method and thesensitivity method are employed to quantify the impacts of climatechange and human activities on runoff. The developed method isapplied to the Weihe River, and the main findings are as follows:

C. Jiang et al. / Journal of Hydrology 522 (2015) 326–338 337

(1) The results of the trend analysis of the annual hydro-meteo-rological series during 1960–2009 indicate that the precipi-tation and runoff display decreasing trends at differentsignificance levels, while the temperature series present asignificant increasing trend, and the potential evaporationhas no significant trend. In addition, it has been also foundthat the annual parameter w in each Budyko-type equationpresents significant increasing trends.

(2) According to the covariate analysis, w in each Budyko-typeequation is related to the explanatory variables of both cli-mate conditions and human activities. Representing param-eter w as a function of variables of both climate conditionsand human activities improves the performance of simulat-ing the mean annual runoff based on all the Budyko-typeequations considered in this paper.

(3) The decomposition method and the sensitivity method areused to separate the contributions of climate change andhuman activities to the decreased runoff in the Weihe Riverbased on the covariate analysis of w. Both methods indicatethat climate change or human activities have differentimpacts on runoff of the Weihe River in different periods,while climate change plays the major role on leading tothe decline of the runoff in most periods. The impacts of cli-mate change on runoff are not only due to the direct changeof the hydrological inputs to the watershed, but also throughits impacts on the watershed characteristics as representedby the parameter w in each Budyko-type equation. In addi-tion, human activities have significantly decreased runoffof the Weihe River via altering the value of w.

Acknowledgements

This research is supported by the National Natural ScienceFoundation of China (Grant Nos. 51190094, 51479139), whichare greatly appreciated. Great thanks are due to the reviewers fortheir constructive comments and suggestions that have greatlyhelped the improvement of the manuscript.

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