Detecting the dominant contributions of runoff variance across the source region of the Yellow River using a new decomposition framework

Quantifying the contributions of climatic variables to runoff variance is still a great challenge to water resource management. This study adopted an extended Budyko framework to investigate the effects of terrestrial water storage changes (ΔS) on runoff variance across the source region of the Yellow River, China, during the period of 2003–2014. A new decomposition framework based on the extended Budyko framework was proposed to effectively quantify the contributions of different climatic variables including precipitation, PET and ΔS to runoff variance. The results demonstrated that the extended Budyko framework showed a better performance in presenting the water and energy balance than the original Budyko framework, especially at fine time scales. Meanwhile, the variance in runoff estimated by the new decomposition framework was close to that of runoff observations, indicating that this framework can effectively capture the variation in runoff during 2003–2014. It was also found that precipitation was the most important factor that contributed to runoff changes, while PET made a slightly smaller contribution compared to precipitation. Notably, the results also emphasized the important effects of ΔS on runoff variance at fine time scales, which was useful to better understand the interactions between atmospheric and hydrological processes for regions.


INTRODUCTION
Runoff is an important source of natural freshwater resources, which has been regarded as the fundamental element in social and economic development (Tikhamarine et al. ). Understanding the main drivers of runoff changes and variability is essential to decision-making in regional freshwater resource planning and management (Berghuijs et al. ). During the past decades, impacts of climatic variability on runoff magnitude have attracted widespread attention from many hydrologists and water resource managers. It is reported that about 31% of 145 major rivers across the world have shown significant changes in the mean annual runoff during the past decades (Walling & Fang ; Zhai & Tao ). Therefore, a comprehensive and quantitative understanding of runoff variance and its main sources is totally necessary for the evaluation of hydrological models (Wang & Hejazi ), climatic variability impact assessment (Collins et al. ) and uncertainty quantification in runoff estimation (Bock et al. ). () estimated the sensitivity of runoff to the mean annual precipitation across the world based on an elasticity indicator and concluded that annual changes in precipitation were amplified in runoff, that is, a 1% change in the mean annual precipitation resulted in a 1-3% change in the mean annual streamflow globally.
Aside from various attribution methodologies mentioned above, the water balance method in combination with the Budyko framework has been considered as another useful approach to assess the contributions of different climatic variables and basin characteristics to runoff changes. The Budyko framework has been widely used in previous studies because it can accurately reflect the balance between water availability and energy supply across the study regions (Liang et  season or month) is also rarely made.
The source region of the Yellow River (hereafter SRYR), supplying water for millions of people, is an important region in China. Identifying and analyzing the variations in runoff across the SRYR is useful not only for understanding the water and energy balance between atmospheric and land surface hydrologic processes but also for improving the water resource management across the entire Yellow River basin (Xu et al. ). In addition, the mechanism behind runoff changes in the SRYR has attracted intensive concern in the decade owing to its link to the 'Three Rivers Source Region Reserve' and 'Grain for Green Project'. Several studies have shown that the SRYR has experienced a significant decrease in runoff in recent years (Wang et al. ; Chu et al. ), which will further lead to variations in the temporal and spatial distributions of the water resources in the whole Yellow River basin. However, it has long been challenging to accurately assess the contributions of various natural drivers to runoff variance for this region due to its harsh environment and complex topography. Therefore, it is necessary and meaningful to quantify the contributions of various climatic variables to runoff, which will not only provide valuable guidance for decision-makers in making appropriate policies for water resource assessment but can also help hydrologists better understand runoff response to climatic variability.
As mentioned above, enough knowledge of the temporal variability of runoff is imperative to a comprehensive understanding of hydrological processes under climatic variability.
Although previous studies have consistently investigated the effects of climatic variability on runoff to some extent, they fail to accurately quantify the contributions of these factors to runoff variance (Zarghami et al. ; Li et al. ; Lei et al. ). The main objective of this study is, therefore, to investigate the applicability of the extended Budyko framework by incorporating the effects of terrestrial water storage changes (hereafter ΔS) on regional water and energy balance at different time scales. Meanwhile, a new variance decomposition framework is developed to quantify the main factors influencing the runoff for the SRYR at intra-annual and inter-annual time scales, respectively. To our knowledge, current studies about the impacts of ΔS and other factors on runoff variance at fine time scales are still rare. Therefore, this study will be beneficial for accurately quantifying the response of hydrological cycles to climatic variability, especially for regions with limited hydrological data.
The other parts of this paper are mainly structured as follows. In the 'Study area' section, the study area is briefly described. The 'Data' and 'Methods' sections present the data and methods used in this study, respectively. The 'Results' section contains all the results illustrating the water balances of basins based on the extended Budyko framework at different time scales. Meanwhile, the effects of different factors on variations in runoff are also included in this section. Finally, all the discussions and major conclusions are presented in the 'Discussions' and 'Conclusions' sections, respectively.

STUDY AREA
The SRYR mainly refers to the region controlled by the Tangnaihai (hereafter TNH) hydrological station located in the mainstream of the Yellow River, which is the second-longest river in China with a length of over 5,400 km (shown in Figure 1). The SRYR has a drainage area of 1.22 × 10 5 km 2 and accounts for approximately 15.2% of the area of the entire Yellow River basin. This region is situated in the northeastern Tibetan Plateau, which has an average altitude of over 4,000 meters above sea level. Due to its extremely harsh environment and climatic conditions, the SRYR has long been a sparsely populated region, and therefore this region can be viewed as a relatively pristine area with few  http://data.cma.cn/). All the above data are jointly used to obtain daily time series of potential evapotranspiration (PET) across the study region. In addition, the Thiessen polygon method is adopted to estimate daily time series of precipitation (P) across the study region using observations from different meteorological stations (shown in Figure 1), which are further aggregated to monthly results. Note that some monthly values of GRACE data are missing because of 'battery management' during the study period.
Therefore, these missing values are linearly interpolated from the previous and following months of the corresponding missing data. More detailed descriptions of datasets that are used in this study can be found in Table 1.

Penman-Monteith method
In this study, daily time series of potential evapotranspiration are calculated based on the Penman-Monteith equation, which is viewed as one of the most optimal methods to reflect the energy availability for regions where PET is the potential evapotranspiration (mm day À1 ); Δ is the slope of the saturated vapor pressuretemperature curve (kPa C À1 ); R n is net radiation at the canopy surface (MJ m À2 day À1 ); G is the soil heat flux (MJ m À2 day À1 ); γ is a psychrometric constant (kPa C À1 ); T is the mean air temperature at a height of 2 m ), which can be described as follows: where ΔS is terrestrial water storage changes (mm); P is precipitation (mm); R is runoff (mm); ET is evapotranspiration (mm), which can be estimated by the average of three different evapotranspiration products as mentioned in the 'Evapotranspiration' section.
Monthly time series of ΔS across the SRYR are obtained based on the water balance method (Equation (2)). To further validate the accuracy of these results derived from the water balance method, the changes in terrestrial water storage across the SRYR are also estimated based on  GRACE data independently: where ΔS 0 is GRACE-derived terrestrial water storage changes (mm); TWSA(m þ 1) is terrestrial water storage anomalies (mm) for month (m þ 1), while TWSA(m À 1) is terrestrial water storage anomalies (mm) for month (m À 1).
Previous studies ( where ∅ ¼ PET p is the long-term average aridity index for study regions; P and PET are precipitation (mm) and potential evapotranspiration (mm), respectively; n is the parameter reflecting basin-specific characteristics, such as soil moisture, vegetation cover and climate seasonality where ∅ (¼PET/P) and n represent the aridity index and the basin characteristics parameter at different time scales such as annual or monthly.
Although it can capture the annual water energy and water balances for some river basins, the original Budyko it is very necessary to take the influences of ΔS into consideration, which can be obtained from Equation (2). Chen et al.
() suggested replacing atmospheric water supply (P) by the total available water (P 0 ) in Equation (5), and therefore the original Budyko framework can be extended into: where P 0 (¼P À ΔS) is effective precipitation (mm), which represents the total available water for regions and mainly depends on both atmospheric water supply and basin storage (Wu et al. a,); ∅ 0 (¼PET/P 0 ) represents the aridity index considering ΔS; and n is the basin characteristics parameter, which can be obtained based on the least squares method.

Runoff variance decomposition framework
To further quantify the effects of climatic factors (including P and PET) and basin storage changes (i.e. ΔS) on runoff, a new variance decomposition framework is proposed in this study. According to the water balance equation and assuming that the long-term storage change ΔS ¼ 0, the observed runoff (R) deviation from its long-term mean ( R) at a specific time interval can be expressed as follows: where ΔR i , ΔP i , ΔET i and ΔS i refer to the changes in runoff, precipitation, evapotranspiration and terrestrial water storage at a specific time interval, such as month, season or year.
As suggested by Zeng & Cai (), changes in ET (i.e. ΔET i ) for regions mainly consist of different changes in P, PET and ΔS, respectively, which can be described as: where ΔR i , ΔP i and ΔS i are the same as that shown in Equation (7), ΔPET i represents changes in potential evapotranspiration, ∅ is the long-term average aridity index, F( ∅) and F 0 ( ∅) are the Budyko framework and its firstorder derivative, respectively.
To further substitute the term of ΔET i from Equation (7) into Equation (8) yields: The sample variance of R can therefore be derived by taking the square of Equation (9), summing over N samples and scaled by N À 1: where σ 2 and cov( ) indicate the variance and covariance of different factors, respectively. Additionally, w i before the different variance (or covariance) terms represents the corresponding weighting factors, respectively, which can effectively quantify the contributions of different factors to R variance and can be analytically estimated from the long-term average aridity index ( ∅ ¼ PET = P) in combination with the Budyko framework. Different weighting factors are presented as follows: where is the long-time average aridity index; F( ∅) and F 0 ( ∅) represent the original Budyko framework and its first-order derivative, respectively; and w i refers to different weighting factors. In these equations, a positive (or negative) weighting factor indicates that an increase (or decrease) in the corresponding term (such as σ 2 P , σ 2 R and σ 2 ΔS ) will result in an increase (decrease) in runoff variance (σ 2 R ). For example, if w p is equal to 0.1, this means that a 10% increase of precipitation variance (σ 2 P ) would simultaneously bring a 1% increase in runoff variance (σ 2 R ). According to Equations (10) and (11)  Monthly time series of runoff is also acquired from the TNH station, which shows a reasonable correspondence with the precipitation and ET shown in Figure 2.

Validation of changes in terrestrial water storage across the SRYR
To better understand the variations of water cycles and their response to climatic variability across the SRYR, the ΔS at different time scales (monthly, seasonal and annual, respectively) is needed. In this study, regional ΔS is estimated as the residual of water balance closure. As shown in Figure 3, monthly time series of ΔS based on the water balance method shows a seasonal variation with a wide range from À41.2 to 59.8 mm.
Since the aforementioned results are directly estimated by the water balance method, it is necessary to further vali-    when neglecting the effects of ΔS at the monthly scale. In addition, many data points of PET/P and ET/P obviously depart from the theoretical Budyko curve, and some scatter points even fall above the water limit line of Budyko space (i.e. ET/P >1). This is inconsistent with the previous results found by Chen et al. (), who stated that the obvious linear relationship between ET/P and PET/P can be captured at the monthly scale in some typical arid and semiarid regions.
In contrast, the data points of PET/P 0 and E/P 0 considering ΔS generally follow the theoretical Budyko curve well when considering ΔS for the SRYR at the monthly scale as shown in Figure 4(b). Overall, the points scaled by P 0 (i.e. with considering ΔS, Figure 4(b)) are denser than those scaled by P (i.e. without considering ΔS, Figure 4(a)) at the monthly scale. Additionally, the data points considering ΔS (R 2 ¼ 0.49) show a better performance in the fitting of Budyko curves than those without considering ΔS (R 2 ¼ 0.11), indicating that ΔS plays an important role in the basin-scale water and energy balance. The above results further reflect that regional ΔS indeed makes up a large proportion of the partitioning of P into R (or E), and therefore it should be taken into consideration when analyzing the hydrological cycles at the monthly scale.

Seasonal scales
Investigating the water and energy budget over regions at seasonal scales can help us better understand the variability in runoff, which is important and meaningful for predicting some extreme events such as droughts or floods in advance under climatic variability, especially in some ungauged basins. Therefore, the Budyko framework is also applied in the SRYR using seasonal time series of data during the study period. Similar to the results shown in Figure 4, Figure 5 also presents the water balance in the Budyko curve for the SRYR. The results in Figure 5(a) demonstrate that the data points representing PET/P versus E/P distribute as linear curves rather than the Budyko curves as expected while assuming that ΔS was negligible at the seasonal scale. Some data points exceeded the 'water limit' boundary represented by ET/P ¼ 1 in Figure 5(a), indicating that actual ET is more than the amount of P due to the neglect of ΔS in the water balance equation. considering ΔS for the SRYR. As expected, the data points considering ΔS in the study period present a perfect Budyko relationship at the seasonal time scale as depicted in Figure 5(b). In other words, the extended Budyko framework considering ΔS (i.e. Equation (6)) can accurately capture the seasonal variability of the regional water and energy balance across the SRYR in this figure.

Annual scales
At the annual scale, both the scatter points shown in  In general, using the newly modified effective precipitation (P 0 ¼ P -ΔS) as the proxy of total water availability for closed basins, the data points of PET/P 0 and ET/P 0 (R 2 ¼ 0.81) generally show a better performance in the fitting of the Budyko relationship than those without considering ΔS (R 2 ¼ 0.67). In particular, the original Budyko framework using the precipitation as the available water resource is reasonable and applicable for the longterm hydrological cycles of natural and closed catchments because ΔS is relatively small compared to variations in runoff and the other variables such as P or ET at the  annual scale. When the time scale becomes finer, the contribution of ΔS to the water balance will become bigger, which indicates that the effects of ΔS are generally significant in the SRYR at the monthly and seasonal scale.

Contributions of climate factors and basin storage to
runoff variance based on the extended Budyko framework It is widely known that runoff is closely related to regional water resource management and planning, which is of critical importance for sustainable social and economic development (Mccabe & Wolock ; Fowler et al. ). In fact, variation in runoff is usually viewed as one of the most important indicators that can reflect the true state of regional available water. Therefore, deep insights into runoff variance can help us better assess how regional water availability (in the form of runoff) has changed in the past decade. According to the newly proposed variance decomposition framework (i.e. Equation (10) (10)) are shown in Figure 7, which represents the corresponding contributions to runoff variance when the same changes occurred in each term. It can be found that the value of w P is identical to that of w ΔS shown in Equation (10), which indicates that P and ΔS can make same contributions to runoff variance on the condition that same variations occur in P and ΔS, respectively. Additionally, the negative contribution from the covariance between P and ΔS (i.e. w P, ΔS) to runoff variance is the most significant among all weighting factors with a value of À0.43. In comparison with the other weighting factors listed in Figure 7 (10), such as w P,PET cov(P, PET ) and w P,PET cov(P, ΔS), generally make obviously negative contributions to runoff variance, which indicate that the source of runoff reduction mainly arises from the covariance between P and PET (or ΔS). In addition, variance in runoff estimated by Equation (10) (σ 2 R sim) has a good agreement with that derived from observations (σ 2 R obs) at the monthly scale with a small bias of 7%, which demonstrates that the variance decomposition framework is applicable and effective to assess the contributions of terms to runoff variance across the SRYR. In contrast, the negative contribution to the runoff variance results from the terms, including w P,PET cov(P, PET ) and w P,PET cov(P, ΔS), which is similar to that shown in Figure 8(a). Additionally, variance in runoff predicted by Equation (10) (σ 2 R sim) is 847 mm 2 , which is rather close to the results derived from observations (σ 2 R obs) at the seasonal scale (i.e. 736 mm 2 ) with a relative bias of 15%. According to the proposed variance decomposition framework, the contributions of different climate factors and basin storage to runoff variance for the SRYR have been fully quantified. Overall, P contributes more to runoff variance than the other factors such as PET and ΔS because it is the most important input for regional water balance. Especially, this study emphasizes the effect of ΔS on runoff variance. As mentioned in Figure 8, the effects of ΔS on runoff variance mainly come from three parts, that is, ΔS and the related covariance terms including w P,PET cov(P, PET ) and w P,PET cov(P, ΔS). It should be noted that ΔS can contribute more to the variance in runoff than PET. Although the effects of ΔS have long been neglected in previous studies (Jiang et al. ; Wang et al. ), the results from this study have demonstrated that ΔS may play a critical role in runoff variance.
Obviously, the runoff variance simulated by Equation (10) would be possibly underestimated or overestimated at any time scales without considering the effect of ΔS.

Performance of Budyko framework at different time scales
To investigate the role of ΔS (including variations in surface water storage, soil moisture storage and groundwater storage) in the water balance of the SRYR, the variation in terrestrial water storage has been firstly estimated, which shows good agreement with that derived from GRACE data. ΔS and other factors including P and PET were jointly used to analyze the regional water and energy balance at different time scales based on the Budyko framework as mentioned in the 'Results' section.
Overall, the results demonstrate that it is reasonable to exclude the variations in terrestrial water storage from precipitation (i.e. P 0 ¼ P À ΔS), which is more consistent with the water availability concept in boundary conditions of the original Budyko framework, especially at finer time scales such as monthly or seasonal (Chen et al. ). Meanwhile, this result repeatedly shows the importance of ΔS when analyzing the regional water balance, since the water available estimation by using P only is highly overestimated. Similar results also have been obtained in some other regions around the world (Zeng & Cai ).

Main factors controlling runoff variance at different time scales
The variance decomposition framework of the runoff was applied at monthly, seasonal and annual scales independently. The results show that the runoff variance simulated by the decomposition framework is in line with that observed by the hydrological station in general. However, there still exists some discrepancy between the simulated runoff variance and that derived from observations, especially at the annual scale. This phenomenon can be explained by the following two factors. On one hand, precipitation is viewed as the sole water input for hydrological cycles in the study region when estimating ΔS based on the water balance method, which has proved to be reliable and robust across the SRYR. However, some other natural water resources, such as the melting of snow or glaciers, are not taken into consideration due to the lack of in situ observations. On the other hand, the term of aridity index (i.e. ∅) in Equations (11)

Uncertainties and limitations
In this study, different terms in the water balance method are realistically derived from the meteorological stations.
In fact, it remains a big challenge to obtain accurate and reliable estimations about precipitation or PET in highaltitude regions such as the SRYR. Furthermore, extremely limited meteorological stations also make it more difficult to capture the true state of water and energy balance cycles in this region. ET, as one of the most difficult variables to obtain or measure, at regional and basin scales, is difficult to be accurately estimated due to its link to Earth's water, energy and carbon cycles (Wang et al. ). Therefore, the average of three widely used ET products is adopted to describe the variations in ET with the goal of reducing error and uncertainty. However, the uncertainty of different terms mentioned above will further lead to an error in ΔS through the water balance method (i.e. Equation (2)) via the principle of uncertainty propa- The newly proposed variance decomposition framework is physically robust, while some high-order terms in this equation have been neglected in this study for simplicity. Although the errors induced by these high-order terms are relatively limited, it may still lead to some discrepancies between the estimated and theoretical results. Given the above reasons, runoff variance estimated by the proposed variance decomposition framework would be possibly overestimated or underestimated. Therefore, more efforts will be made in our next study to reduce these errors and obtain more reliable results.

CONCLUSIONS
This study applied the original and extended Budyko framework to investigate the role of ΔS in water and energy balance across the SRYR at different time scales. A new variance decomposition framework was also proposed to partition the variance in runoff into different climate factors and basin storage. The major findings from this study are summarized as follows: (1) Variations in terrestrial water storage have proved to play an important role in the hydrological cycles, and neglecting the effects of ΔS would result in obvious errors when applying the original Budyko framework to analyze the water and energy balance for regions, especially at fine time scales such as month or season.
In comparison with the original Budyko framework, the extended Budyko framework considering ΔS can better reflect the true process of exchange between water and energy across the study region.
(2) The variance decomposition framework proposed in this study can be applied to effectively assess the contributions of different climatic factors and basin storage to the variance in runoff at different time scales. It was found that this method is more effective at monthly or seasonal scales than at the annual scale, which may result from the limitation of sample data available at annual scales.
(3) According to the proposed variance decomposition framework, P is the main source of the variance in runoff for the SRYR and has significantly positive effects on variance in runoff. ΔS makes a positive contribution to variance in runoff, which is more significant at monthly and seasonal scales than the annual scale. In addition, the different covariance terms between P and PET (or ΔS) can have a significant influence on variance in runoff.
The conclusions drawn from this study may help us better understand the response of hydrologic cycles under climatic variability across the SRYR, which can provide valuable guidance for decision-makers in evaluating and predicting water resources. According to the extended Budyko framework, the variance decomposition framework proposed in this study can make a quantitative assessment of the influence of climatic variability on runoff changes, which can be applied in other study regions.

AUTHOR CONTRIBUTIONS
Y.-P.X. and J.X. designed the study; J.X. did the main calculations and wrote the draft of the manuscript; Y.-P.X. guided the research and revised the manuscript; Y.W. and Y.G. performed data preprocessing.

CONFLICT OF INTEREST
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.