Estimation and attribution of water storage changes in regulated lakes based on Budyko's supply–demand framework

doi:10.21203/rs.3.rs-1632974/v1

Understanding changes in lake water storage (LWS) and its attribution to climate change and human activities is essential for adaptive management of water resources in regulated lakes. Although altimetric measurements, whether ground- or satellite-based, can reproduce LWS dynamics, they do not provide sufficient information on why LWS is changing. Neither the water balance method nor detailed hydrological modeling can capture exactly LWS change in most regulated lakes largely because of the scarcity of water output observations. Here, a simple water balance model, following Budyko’s supply–demand framework, was proposed to estimate LWS change in regulated lakes without the need for water output information. The elasticity of LWS was theoretically derived from the Budyko-based model to attribute LWS change to its main driving factors. The annual LWS observed during 1964–2019 under three different regulation plans in Lake Dianchi, which is a typical regulated lake located in southwest China, was used to calibrate and test the proposed model. Comparison between the estimated results and observed data indicated that the model accurately captured the variations of annual LWS in Lake Dianchi with a relative root mean square error of 4.08%, and mean absolute percentage error of 2.88%. The attribution results suggested that lake regulation strategies were the primary cause of LWS change in Lake Dianchi, while water transfer had a limited contribution. This study suggests that complex hydrological behavior in regulated lakes can be explored using Budyko's supply–demand framework with a low requirement for data. This can provide effective guidance to lake managers and policymakers for adaptive management of water resources in regulated lakes.

Lake water storage

Budyko framework

Regulated lake

Attribution analysis

Lake Dianchi

Lake water storage (LWS) is one of the most important geometric characteristics of natural lakes, and determines their ecological functions and socioeconomic values (Boehrer et al. 2010). Globally, LWS accounts for nearly three quarters of the readily accessible water resources on Earth (Molinos et al. 2015; Messager et al. 2016), and plays an essential role in ensuring water security and supporting water-related ecosystem services (Verpoorter et al. 2014; Mathis et al. 2016). However, LWS is highly sensitive to changes in regional environments because it incorporates the impacts of all natural and anthropogenic disturbances on the hydrological budget (Haghighi and Kløve 2015; Guo et al. 2015). Owing to the combined impacts of climate change and intensified human activities in recent decades, LWS is undergoing significant changes around the world, causing an increased risk to water security, degradation of lake-related ecosystems, and mismanagement of water resources (Gibson, et al. 2006a; Song et al. 2014; Pekel et al. 2016; Yang et al. 2016; Liu et al. 2019). Therefore, a better understanding of the response of LWS to natural and human-induced disturbances is extremely important for lake managers and policymakers.

In sparsely populated regions, LWS varies because of the imbalance between water inflows and outflows resulting from natural variability (Molinos et al. 2015). Therefore, long-term LWS change in near-natural lakes solely reflect the effect of climatic change occurring in the region (Song et al. 2014a). Several studies have been conducted to investigate the response of LWS to natural drivers such as precipitation, evapotranspiration and glacier shrinkage (Song et al. 2014b; Fan et al. 2021). Compared with that in sparsely populated regions, LWS in densely populated regions reflects the additional impacts from human activities, including water regulation. To better serve human needs, natural lakes in inhabited regions are mostly regulated by engineered control structures (e.g., levees, dams and sluices). Notable examples worldwide include Lake Ontario in America (Wilcox and Xie 2007), Lake Baikal in Asia (Jaguś et al. 2014), and Lake Victoria in Africa (Cetirana et al. 2020). Existing studies show that the operation of engineered control structures has a significant effect on LWS dynamics by regulating lake outflows and dampening water level variability (Gibson et al. 2006b; Yin et al. 2013; Getirana et al. 2020). The water balance in regulated lakes is also subject to other human-induced disturbances (e.g., within-lake ecological restoration, inter-basin water transfer, irrigation and other withdrawals), which complicate the operational schemes for lake regulation and alter the variations in LWS (Veijalainen et al. 2010; Fergus et al. 2019; Getirana et al. 2020). The complicated hydrological behavior of regulated lakes brings challenges to understanding LWS dynamics and their response to the changing environment.

Although LWS changes can be estimated from the differences in the water level recorded at hydrometric stations (Yin et al. 2013; Xu et al. 2020) or monitored from satellite-based platforms (Zhang et al. 2006; Swenson and Wahr 2009; Han et al. 2020), these observational records alone may not provide sufficient information on why LWS is changing in regulated lakes (Lei et al. 2014). So far, previous efforts to address this issue were mostly made by means of water balance analysis, which derives LWS change as the difference between the water inputs and outputs of the lake (Chebud and Melesse 2009; Veijalainen et al. 2010; Hassan and Jin 2014; Zhang et al. 2019). As the closure term of the balance, an exact estimation of LWS change relies on accurate information on all the water input and output terms (Gronewold et al. 2020). However, these data are not available for most regulated lakes, especially for water output terms. Differ from these Newtonian methods with focusing on the physical processes in isolation, the Darwinian methods explain systematically the hydrological behavior as a whole, and propose an alternative way to capture the relevant processes through simple but effective empirical functions (Wang and Tang 2014). The most popular among these is Budyko hypothesis. Although the Budyko hypothesis was originally proposed to partition the annual water balance of natural, closed basins as a function of the ratio of available energy (the demand) to water (the supply) (Budyko 1958), recent efforts have generalized it as a supply–demand framework to partition available water into several components (Zhang et al. 2008; Wang, et al. 2011). Owing to its simplicity and ability to reflect complex hydrological processes with low data requirements, the generalized Budyko framework has been applied to explain various water-related processes, such as human water consumption (Lei et al. 2018), infiltration (Wang 2018), and baseflow (Cheng et al. 2020). Moreover, Budyko framework has been widely used to attribution analysis of hydrological components (e.g., runoff, soil water storage, and evapotranspiration) in response to climatic change and human activities (Wang and Hejazi 2011; Yang et al. 2020; Krajewski et al. 2021; Li et al. 2022). Similarly, there has been a potential need to develop such a Budyko-type model for assessing LWS changes in regulated lakes.

Actually, when the focus is on changes in LWS, it is not necessary to acquire detailed information on all output terms and these can be replaced by a total water output term. Hence, the available water in a lake can be partitioned into actual LWS and total water output. For regulated lakes, two operational water levels are typically specified in the regulation rules: maximum controllable and minimum drawdown (Parisopoulos et al. 2009). Therefore, apart from the constraint of the available water in the lake, actual LWS is also bounded by the effective storage capacity (i.e., the potential retention in the lake), which is controlled by the two water level limits for lake regulation. These two constraints on water balance in regulated lakes are similar to the water and energy constraints on catchment water balance (Budyko 1958). In a similar fashion to the Budyko-based estimation of evapotranspiration by partitioning precipitation without the need for streamflow information, there may be a potential alternative to LWS estimation through a Budyko-based partitioning of available water in regulated lakes without the need for water output information. Furthermore, the proposed Budyko model can be applied for sensitivity analysis to identify the main driving factors, and quantify their contribution to LWS change.

The objective of this study was (1) to propose a Budyko-based model that could estimate LWS change without prior information on water outputs; (2) to select a typical regulated lake to test the proposed model; and (3) to apply the elasticity method derived from the Budyko-based model to quantify the contribution of the main driving factors to LWS change. The reminder of this paper is organized as follows: the next section introduces the study area and data sources, Section 3 proposes the methods used in this paper, Section 4 presents the results and discussion, and the main conclusions are summarized in Section 5.

2.1 Study area

The study site was the Lake Dianchi catchment with a total area of 2920 km², located in central Yunnan, southwest China (Fig. 1). The catchment is a semi-closed lake basin with only one natural outlet (the Haikou River) which drains into the Yangtze River. The catchment has a mountainous sub-tropical moist monsoon climate, with average annual temperature of 15.2°C, precipitation of 993 mm, and pan evaporation of 1211 mm. Almost one-tenth of the catchment is covered by Lake Dianchi, China’s sixth-largest freshwater lake (Wang 1999), which has a surface area of 309 km². Located at the lowest point of the catchment, the lake receives water inflows from more than 30 centripetal rivers. The main inflowing tributaries are the Panlong River, Baoxiang River and Chai River.

<Fig. 1>

Although the area around the lake has been inhabited for approximately 2500 years by the ancestors of the Dian tribe, Lake Dianchi had a natural outflow in all historical periods until 1836 when the Lyfeng Sluice Gate was built at its outlet. Since then, the outflow of the lake has been regulated to facilitate the yearly dredging of the outflow river. In 1956, to mitigate the increasing conflict between flood control and water supply, the local government rebuilt the Lvfeng Gate and formulated an operational plan for lake regulation (hereafter referred to as Plan-1956). Under Plan-1956, the water level of Lake Dianchi was regulated within the range of 1887.2 m (upper limit) and 1885.5 m (lower limit). The inflow of excessive untreated sewage into the lake from accelerated population growth and increased demand for food production led to a rapid deterioration in water quality and severe eutrophication in 1980s (Liu et al. 2014). As a result, eco-environmental restoration became the primary objective of lake regulation, and local government enacted a revised regulation plan to incorporate the need for lake protection in 1988 (termed as Plan-1988). The upper limit of the water level was raised to 1987.4 m. Later in 1996, the lake was divided into two parts by an artificial dam with an added artificial outlet (Xiyuan Tunnel), which was built to increase the outflow capacity. In 2012, the regulation plan was further revised to optimize water quality improvement and the upper limit of the water level was raised to 1987.5 m (termed as Plan-2012). Furthermore, in 2013 the Niulanjiang-Dianchi Water Transfer Project (NDWTP) was built to transfer an annual mean 572 million m³ of freshwater from outside the basin into Lake Dianchi to dilute the polluted water (Dai et al. 2016). Under these three different regulation plans, LWS estimated from water level records showed a significant increase during 1964–2019 (Fig. 2). For the adaptive management of water resources in the regulated lake, there is a pressing need to better understand the impacts of human activities such as lake regulation and inter-basin water transfer on LWS change.

2.2 Data sources

Long-term (1964–2019) monthly precipitation and other meteorological parameters (including precipitation, air temperature, wind speed, relative humidity, and sunshine duration) were collected from three meteorological stations, which are run by the Yunnan Meteorological Service (http://www.ynmb.cn). The water level records observed at two gauging stations, i.e., Haigeng (No. 17 in Fig. 1) and Haikou (No. 18 in Fig. 1), during 1964–2019 by the Yunnan Hydrological Bureau were used to represent the lake water levels. The water level–storage rating curve of Lake Dianchi was surveyed by the Kunming Bureau of Environmental Protection. Additionally, to simulate the lake inflow generated uniformly over the contributing basin, monthly precipitation data from 16 rain gauge stations within Lake Dianchi watershed were collected from Yunnan Hydrological Bureau. More detailed information about these stations can be found in Table 1.

Table 1

Hydro-meteorological data used in this study
Station No.	Station name	Altitude (m asl)	Type	Time span	Resolution	Source^*
1	Ardalong	2180	Precipitation	1964–2019	monthly	YHB
2	Arziying	2100	Precipitation	1964–2019	monthly	YHB
3	Dashiba	2010	Precipitation	1964–2019	monthly	YHB
4	Zhonghe	2003	Precipitation	1964–2019	monthly	YHB
5	Songhuaba	1913	Precipitation	1964–2019	monthly	YHB
6	Huatingsi	2072	Precipitation	1964–2019	monthly	YHB
7	Sanjiacun	2200	Precipitation	1964–2019	monthly	YHB
8	Xibeishahe	1960	Precipitation	1964–2019	monthly	YHB
9	Dongbaishahe	1920	Precipitation	1964–2019	monthly	YHB
10	Dabanqiao	1965	Precipitation	1964–2019	monthly	YHB
11	Baoxianghe	2060	Precipitation	1964–2019	monthly	YHB
12	Hengchong	2003	Precipitation	1964–2019	monthly	YHB
13	Liangwangs	2810	Precipitation	1964–2019	monthly	YHB
14	Shuanglongwan	1923	Precipitation	1964–2019	monthly	YHB
15	Chaihe	1965	Precipitation	1964–2019	monthly	YHB
16	Shuanglong	1920	Precipitation	1964–2019	monthly	YHB
17	Haigeng	1888	Lake level	1964–2019	monthly	YHB
18	Haikou	1889	Lake level	1964–2019	monthly	YHB
19	Chenggong	1907	Meteorological parameters	1964–2019	monthly	YMS
20	Jinning	1891	Meteorological parameters	1964–2019	monthly	YMS
21	Kunming	1891	Meteorological parameters	1964–2019	monthly	YMS
Note: * YHB represents Yunnan Hydrological Bureau, YMS the Yunnan Meteorological Service.

The framework for this study is presented in Fig. 3. First, a Budyko-based lake water balance model was established to estimate LWS change in regulated lakes without the need for water output information. Second, the Shuffled Complex Evolution algorithm (SCE-UA) was used to calibrate model parameters. Third, the model performance and uncertainty of the simulatioactive LWS.n results were evaluated. Finally, quantitative attribution of LWS change was analyzed using the Budyko-based elasticity method.

3.1 Budyko-based LWS change estimation model

3.1.1 Lake water balance equation

The catchment of a lake can be conceptualized as a system of two regions: the lake surface and its contributing basin. The water balance of the lake can be written as (Haghighi and Kløve 2015)

$$S_{l}^{{t - 1}}+\left( {P_{l}^{t} - E_{l}^{t}} \right)A_{l}^{t}+R_{b}^{t}A_{b}^{t}+I_{c}^{t}=S_{l}^{t}+O_{r}^{t}+O_{g}^{t}+O_{h}^{t}$$

1

where S_l is the LWS, P_l is the areal precipitation over the lake, E_l is the evaporation over the lake, R_b is the contributing basin runoff, A_b is the contributing area of the basin, A_l is the surface area of the lake, I_c is the channel inflow from outside the basin, O_r is the outflow through the lake-outlet rivers, O_g is the groundwater outflow, O_h is the human-induced water output associated with water diversion and withdrawal from the lake, and the superscripts t and (t-1) are used to denote the parameters in periods t and (t-1), respectively.

The most common strategy to estimate LWS change from the water balance relies on term-by-term estimation of the components comprising the above equation. On the left-hand side of Eq. (1), the areal precipitation over the lake P_l can be calculated from precipitation data gathered at the meteorological and/or rain gauge stations close to the lake using the Thiessen polygon method (Gibson et al. 2006b). Evaporation from the lake water surface E_l is estimated by the standard Penman-Monteith formulation (Allen et al. 1998). Basin runoff R_b is simulated by the dynamic water balance model (DWBM) (see subsection 3.1.4 for more details). Channel inflow from outside the basin I_c is usually measured. However, it is impossible to accurately estimate the three water output terms on the right-hand side of Eq. (1) for most regulated lakes. For example, the outflow of regulated lakes can be estimated from the operation curve for regulation (Veijalainen et al. 2010; Han et al. 2021). However, actual outflow has been shown to deviate from the estimated results because lake regulation schemes needed to be corrected in real time (Lee et al. 1994; Parisopoulos et al. 2009). Therefore, the equation for estimating LWS change is not always closed.

3.1.2 Rearrangement of the lake water balance equation

For regulated lakes, there are two typical water levels in the regulation rules: maximum controllable h_max and minimum drawdown h_min (Parisopoulos et al. 2009). As illustrated in Fig. 4, h_max is set to the upper limit of the water level for flood prevention purpose, while h_min is commonly set to the lower limit of the water level for lake regulation without damage to the lake ecosystem (Veijalainen et al. 2010). Hence, of the water stored in the lake in period t, the portion above h_min is active (defined as active LWS), while the portion below h_min is static (static LWS). ThereforeEstimation of basin runoff, LWS may be determined by

$$S_{l}^{t}=S_{{la}}^{t}+S_{{ls}}^{t}$$

2

where S_la is the active LWS, and S_ls is the static LWS. To ensure sustainable use, water use should not exceed the active LWS.

Moreover, when focused on the change in LWS, detailedEstimation of basin runoff simulation of the three output terms is not necessary, and thus they can be replaced by a total water output term, such that

$$O_{0}^{t}=O_{r}^{t}+O_{g}^{t}+O_{h}^{t}$$

3

where O₀ is the total water output. For the case with a fixed h_min, the change in static LWS due to sedimentation can be negligible. Hence, substituting Eqs. (2) and (3) into Eq. (1), the lake water balance equation can be simplified as

$$S_{{la}}^{{t{\text{-}}1}}+\left( {P_{l}^{t} - E_{l}^{t}} \right)A_{l}^{t}+R_{b}^{t}A_{b}^{t}{\text{+}}{I_c}=S_{{la}}^{t}+O_{0}^{t}$$

4

From the perspective of water resources, this relationship indicates that water resources contained in the lake consist of the following three parts: naturally replenished, artificially replenished and stored (Izmailova 2018). The first part is water resources replenished through the natural water cycle, and includes tributary inflow, and precipitation minus evaporation from the lake, which is given by

$$W_{n}^{t}=\left( {P_{l}^{t} - E_{l}^{t}} \right)A_{l}^{t}+R_{b}^{t}A_{b}^{t}$$

5

in which W_n is the water resources replenished through the natural water cycle. The second part is water resources transferred from outside the basin through the inter-basin water transfer project, i.e.

$$W_{a}^{t}=I_{c}^{t}$$

6

in which W_a is the artificially replenished water resources. On the left-hand side of Eq. (4), the remaining term is the stored water resources, i.e.,

$$W_{s}^{t}=S_{{la}}^{{t - 1}}$$

7

in which W_s is stored water resources. Therefore, the total usable water resources contained in the lake can be determined by

$$W_{0}^{t}=W_{n}^{t}+W_{a}^{t}+W_{s}^{t}$$

8

in which W₀ is the total usable water resources. Hence, substituting Eqs. (5)~(8) into Eq. (4), the lake water balance equation can be rewritten as

$$W_{0}^{t}=S_{{la}}^{t}+O_{0}^{t}$$

9

The relationship indicates that the total usable water resources in the lake is partitioned into active LWS and total water output. Similar to the Budyko hypothesis for estimating evapotranspiration through a simple partitioning of precipitation without the need for streamflow information, this study proposed a Budyko-based model to estimate active LWS by partitioning the total usable water resources in regulated lakes without the need for water output information.

3.1.3 Budyko-based model for LWS change estimation

The Budyko hypothesis was originally proposed to estimate the evapotranspiration ratio as a function of the ratio of available energy (the demand) to water (the supply) in natural, closed basins (Budyko 1958). More recently, it has been extended to solve various supply–demand balance problems in hydrological modeling on the basis of a supply–demand framework (Zhang et al. 2008; Lei et al. 2018; Simons et al. 2020). The general Fu-type expression for the extended Budyko framework is as follows (Zhang et al. 2008; Wang et al. 2011):

$$\frac{Z}{X}=F\left( {\frac{Y}{X},\alpha } \right)=1+\frac{Y}{X} - {\left[ {1+{{\left( {\frac{Y}{X}} \right)}^\alpha }} \right]^{1/\alpha }}$$

10

in which F is the function symbol, X represents the supply, Y the demand, Z the consumption, and α is the consumption curve parameter with a range of (1, ∞).

To partition the total usable water resources in regulated lakes within Budyko’s supply- demand framework, the “supply” term is the total usable water resources (X=$W_{0}^{t}$), and the “demand” term is the effective storage capacity, which is defined as the storage capacity between the two limits of the water level:

$$Y=S_{e}^{t}=S_{{l*}}^{t} - S_{{ls}}^{t}$$

11

in which S_e is the effective storage capacity of the lake, S_l* is the maximum LWS corresponding to the maximum controllable h_max, as shown in Fig. 4. Therefore, as the “consumption” term, the active LWS (Z=$S_{{la}}^{t}$) can be calculated as follows:

$$\frac{{S_{{la}}^{t}}}{{W_{0}^{t}}}=F\left( {\frac{{S_{e}^{t}}}{{W_{0}^{t}}},{\alpha _1}} \right)=1+\frac{{S_{e}^{t}}}{{W_{0}^{t}}} - {\left[ {1+{{\left( {\frac{{S_{e}^{t}}}{{W_{0}^{t}}}} \right)}^{{\alpha _1}}}} \right]^{1/{\alpha _1}}}$$

12

in which α₁ is lake storage efficiency. The larger the value of α₁, the more water is stored in the lake, and the less is the water output from the lake. When ${\alpha _1} \to \infty$, the term in Eq. (12) ${\left[ {1+{{\left( {{{S_{e}^{t}} \mathord{\left/ {\vphantom {{S_{e}^{t}} {W_{0}^{t}}}} \right. \kern-0pt} {W_{0}^{t}}}} \right)}^{{\alpha _1}}}} \right]^{1/{\alpha _1}}} \to 1$, then one has $S_{{la}}^{t}=S_{e}^{t}$. It indicates that the active LWS (the consumption) perfectly matches the effective storage capacity of the lake (the demand). In reality, the demand is often unmet, the active LWS is below both the effective storage capacity and the total usable water resources, as shown in Fig. 5. Then, the LWS change in period t can be estimated by

$$\Delta S_{l}^{t}=S_{{la}}^{t} - S_{{la}}^{{t - 1}}$$

13

in which $\Delta {S_l}$ is the change in LWS.

3.1.4 Estimation of basin runoff

The DWBM proposed by Zhang et al. (2008) was applied to estimate basin runoff in this study. In the DWBM, the total precipitation over the basin $P_{b}^{t}$ is partitioned into direct surface runoff $R_{d}^{t}$ and catchment water retention $\varPhi _{b}^{t}$, which consists of soil storage change $\Delta S_{b}^{t}$, basin evapotranspiration $E_{b}^{t}$, and ground water recharge $R_{g}^{t}$, as shown in Fig. 4. Considering the basin precipitation as the “supply” term in Eq. (10) (i.e., X=$P_{b}^{t}$), the potential basin water retention as the “demand” term (Y=$\varPhi _{{{b^*}}}^{t}$), and the actual water retention over the basin as the “consumption” term (Z=$\varPhi _{b}^{t}$), the direct surface runoff is determined as follows:

$$R_{b}^{t}=P_{b}^{t}\left( {1 - \frac{{\varPhi _{b}^{t}}}{{P_{b}^{t}}}} \right)=P_{b}^{t}\left[ {1 - F\left( {\frac{{\varPhi _{{b*}}^{t}}}{{P_{b}^{t}}},{\alpha _2}} \right)} \right]$$

14

$$\varPhi _{{b*}}^{t}=E_{{b*}}^{t}+{S_{b*}} - S_{b}^{{t - 1}}$$

15

in which P_b is the basin precipitation, E_b* is the potential evapotranspiration of the basin, S_b* is the soil water storage capacity, S_b is the soil water storage. Φ_b* is the potential basin water retention, α₂ is the basin water retention efficiency. Then, in period t the total available water over the basin $U_{b}^{t}$, i.e., the sum of the basin water retention $\varPhi _{b}^{t}$ and the residual soil storage in the previous period $S_{b}^{{t - 1}}$, can be partitioned into the evapotranspiration opportunity $\varTheta _{b}^{t}$ and groundwater recharge $R_{g}^{t}$. Considering the total available water over the basin as the “supply” term in Eq. (10) (i.e., X=$U_{b}^{t}$), the potential evapotranspiration opportunity as the “demand” term (Y=$\varTheta _{{b*}}^{t}$), and actual evapotranspiration opportunity as the “consumption” term (Z=$\varTheta _{b}^{t}$), the groundwater storage is determined by

$$G_{b}^{t}=\left( {1 - k} \right)G_{b}^{{t - 1}}+U_{b}^{t}\left[ {1 - F\left( {\frac{{\varTheta _{{b*}}^{t}}}{{U_{b}^{t}}},{\alpha _3}} \right)} \right]$$

16

$\varTheta _{{b*}}^{t}=E_{{b*}}^{t}+S_{{b*}}^{t}$ , $U_{b}^{t}=\varPhi _{b}^{t}+S_{b}^{{t - 1}}$ (17)

in which α₃ is the evapotranspiration efficiency, G_b is the groundwater storage, and k is the recession constant of groundwater. Treating groundwater storage as a linear reservoir, groundwater inflow into the lake is calculated as follows:

$$R_{g}^{t}=k \cdot G_{b}^{{t - 1}}$$

18

in which R_g is groundwater inflow into the lake.

3.2 Parameter calibration and uncertainty analysis

The proposed model has five parameters, i.e., lake storage efficiency α₁ and four DWBM parameters (α_2, α_3, S_b*, and k), which need to be calibrated against measured data. The annual values of LWS were converted from the observed water level during the study period using the water level–storage rating curve. Then, the SCE-UA was applied to calibrate the parameters of the proposed model. (Duan et al. 1994). The performance of the proposed model was evaluated using two statistical indicators: relative root mean square error (RRMSE), and mean absolute percentage error (MAPE). In the present study, four ratings of model performance were defined for RRMSE: excellent (RRMSE < 10%), good (10% ≤ RRMSE < 20%), common (20% ≤ RRMSE < 30%), and poor (RRMSE > 30%) (Zhou et al. 2020).

In addition, the 95% confidence limits of the total predicted uncertainties for LWS change were calculated using the Generalized Likelihood Uncertainty Estimation (GLUE) method (Beven and Binley 1992). Three summary statistics proposed by Xiong et al. (2009), namely the containing ratio, the average bandwidth, and the average deviation amplitude, were used to assess the uncertainty (Table 2).

Table 2

Statistical indicators used to evaluate the performance of the proposed model
Purpose	Statistical metrics	Equation	Perfect value
Model performance evaluation	Relative root mean square error (RRMSE)	${\text{RRMSE}}=\frac{1}{{{\mu _m}}}\sqrt {\frac{1}{T}\sum\limits_{{t=1}}^{T} {{{\left( {v_{m}^{t} - v_{e}^{t}} \right)}^2}} }$	0
Model performance evaluation	Mean absolute percentage error (MAPE)	${\text{MAPE}}=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left\| {\frac{{v_{e}^{t} - v_{m}^{t}}}{{v_{m}^{t}}}} \right\|}$	0
Uncertainty analysis	Containing Ratio (CR)	${\text{CR}}=\frac{N}{T}$	1
	Average Bandwidth (B)	$B=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left( {v_{{95\% \hbox{max} }}^{t} - v_{{95\% \hbox{min} }}^{t}} \right)}$	0
	Average Deviation Amplitude (D)	$D=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left\| {v_{e}^{t} - \frac{1}{2}\left( {v_{{95\% \hbox{max} }}^{t}+v_{{95\% \hbox{min} }}^{t}} \right)} \right\|}$	0
Note. ${v_m}$, ${v_e}$are measured and estimated values, respectively; ${\mu _m}$is the average value of the measured data; $V_{{95\% \hbox{max} }}^{t}$, $V_{{95\% \hbox{min} }}^{t}$ represent the upper and lower bound values of the 95% confidence level, respectively; T is the length of data series, and N is the number of observed value that fall within the 95% confidence limits.

3.3 Attribution analysis of change in LWS

The elasticity method based on the Budyko framework was used to estimate the quantitative contribution of change in LWS in this study. First, starting with Eq. (2), the total differential of S_l can be expressed as

$$d{S_l}=\frac{{\partial {S_l}}}{{\partial {S_{la}}}}d{S_{la}}+\frac{{\partial {S_l}}}{{\partial {S_{ls}}}}d{S_{ls}}$$

19

The relative change in S_l can be expressed as follows:

$$\frac{{d{S_l}}}{{{S_l}}}=\frac{{\partial {S_l}/{S_l}}}{{\partial {S_{la}}/{S_{la}}}}\frac{{d{S_{la}}}}{{{S_{la}}}}+\frac{{\partial {S_l}/{S_l}}}{{\partial {S_{ls}}/{S_{ls}}}}\frac{{d{S_{ls}}}}{{{S_{ls}}}}$$

20

Similar to the definition of runoff elasticity (Roderick and Farquhar 2011), the elasticity of LWS to arbitrary independent variable x₁ can be calculated as

${\eta _{{x_1}}}=\frac{{\partial {S_l}}}{{\partial {x_1}}}\frac{{{x_1}}}{{{S_l}}}=\frac{{{x_1}}}{{{S_l}}}$ , x₁ = S_la, S_ls (21)

in which η_x is the elasticity of S_l to the independent variable x₁. Hence, Eq. (20) can be rewritten as follows:

$$\frac{{d{S_l}}}{{{S_l}}}={\eta _{{S_{la}}}}\frac{{d{S_{la}}}}{{{S_{la}}}}+{\eta _{_{{{S_{ls}}}}}}\frac{{d{S_{ls}}}}{{{S_{ls}}}}$$

22

where ${\eta _{{S_{la}}}}$, ${\eta _{_{{{S_{ls}}}}}}$ are the elasticities of LWS to active LWS and static LWS, respectively. As shown in Eq. (12), the change in active LWS are induced by changes in W₀, S_e and α₁, and the differential of S_la can be obtained by the following equations:

$$d{S_{la}}=\frac{{\partial {S_{la}}}}{{\partial {W_0}}}d{W_0}+\frac{{\partial {S_{la}}}}{{\partial {S_e}}}d{S_e}+\frac{{\partial {S_{la}}}}{{\partial {\alpha _1}}}d{\alpha _1}$$

23

The relative change in S_la can be expressed as follows:

$$\frac{{d{S_{la}}}}{{{S_{la}}}}=\frac{{\partial {S_{la}}/{S_{la}}}}{{\partial {W_0}/{W_0}}}\frac{{d{W_0}}}{{{W_0}}}+\frac{{\partial {S_{la}}/{S_{la}}}}{{\partial {S_e}/{S_e}}}\frac{{d{S_e}}}{{{S_e}}}+\frac{{\partial {S_{la}}/{S_{la}}}}{{\partial {\alpha _1}/{\alpha _1}}}\frac{{d{\alpha _1}}}{{{\alpha _1}}}$$

24

Similarly, the elasticity of S_la to arbitrary independent variable x₂ can be calculated as

${\varepsilon _{{x_2}}}=\frac{{\partial {S_{la}}}}{{\partial {x_2}}}\frac{{{x_2}}}{{{S_{la}}}}$ , x₂ = W₀, S_e, α₁ (25)

in which ε_x2 is the elasticity of S_la to the independent variable x₂. Then, Eq. (24) can be rewritten as follows:

$$\frac{{d{S_l}}}{{{S_l}}}={\varepsilon _{{W_0}}}\frac{{d{W_0}}}{{{W_0}}}+{\varepsilon _{{S_e}}}\frac{{d{S_e}}}{{{S_e}}}+{\varepsilon _{{\alpha _1}}}\frac{{d{\alpha _1}}}{{{\alpha _1}}}$$

26

Setting $\varphi ={{{S_e}} \mathord{\left/ {\vphantom {{{S_e}} {{W_0}}}} \right. \kern-0pt} {{W_0}}}$, Eq. (12) becomes

$$\frac{{{S_{la}}}}{{{W_0}}}=f\left( {\varphi ,{\alpha _1}} \right)=1+\varphi - {\left( {1+{\varphi ^{{\alpha _1}}}} \right)^{\frac{1}{{{\alpha _1}}}}}$$

27

where f is the function symbol. The elasticity coefficients of S_ls can be calculated as:

${\varepsilon _{{W_0}}}=1 - \frac{{\varphi \cdot f_{\varphi }^{'}}}{{f\left( {\varphi ,{\alpha _1}} \right)}}$ , ${\varepsilon _{{S_e}}}=\frac{{\varphi \cdot f_{\varphi }^{'}}}{{f\left( {\varphi ,{\alpha _1}} \right)}}$, ${\varepsilon _{{\alpha _1}}}=\frac{{{\alpha _1} \cdot f_{{{\alpha _1}}}^{'}}}{{f\left( {\varphi ,{\alpha _1}} \right)}}$ (28)

Starting with Eq. (8), one has the differential of W₀ as follows:

$$d{W_0}=\frac{{\partial {W_0}}}{{\partial {W_n}}}d{W_n}+\frac{{\partial {W_0}}}{{\partial {W_a}}}d{W_a}+\frac{{\partial {W_0}}}{{\partial {W_s}}}d{W_s}$$

29

Defining the elasticities of W₀ to arbitrary independent variable x₃ as

${\xi _{{x_3}}}=\frac{{\partial {W_0}}}{{\partial {x_3}}}\frac{{{x_3}}}{{{W_0}}}=\frac{{{x_3}}}{{{W_0}}}$ , x₃ = W_n, W_a, W_s (30)

The total differential of W₀ can be used to assess the change in the total usable water resources as follows:

$$\frac{{d{W_0}}}{{{W_0}}}={\xi _{{W_n}}}\frac{{d{W_n}}}{{{W_n}}}+{\xi _{{W_a}}}\frac{{d{W_a}}}{{{W_a}}}+{\xi _{{W_s}}}\frac{{d{W_s}}}{{{W_s}}}$$

31

Substituting Eqs. (26) and (31) into Eq. (22) yields

$$\frac{{d{S_l}}}{{{S_l}}}={\eta _{_{{{S_{ls}}}}}}\frac{{d{S_{ls}}}}{{{S_{ls}}}}+{\eta _{{W_n}}}\frac{{d{W_n}}}{{{W_n}}}+{\eta _{{W_a}}}\frac{{d{W_a}}}{{{W_a}}}+{\eta _{{W_s}}}\frac{{d{W_s}}}{{{W_s}}}+{\eta _{{S_e}}}\frac{{d{S_e}}}{{{S_e}}}+{\eta _{{\alpha _1}}}\frac{{d{\alpha _1}}}{{{\alpha _1}}}$$

32

in which the elasticities of S_l to W_n, W_a, W_s, S_e and α₁ are given by

$${\eta _{{W_n}}}={\eta _{{S_{la}}}}{\varepsilon _{{W_0}}}{\xi _{{W_n}}}$$

33a

$${\eta _{{W_a}}}={\eta _{{S_{la}}}}{\varepsilon _{{W_0}}}{\xi _{{W_a}}}$$

33b

$${\eta _{{W_s}}}={\eta _{{S_{la}}}}{\varepsilon _{{W_0}}}{\xi _{{W_s}}}$$

33c

$${\eta _{{S_e}}}={\eta _{{S_{la}}}}{\varepsilon _{{S_e}}}$$

33d

$${\eta _{{\alpha _1}}}={\eta _{{S_{la}}}}{\varepsilon _{{\alpha _1}}}$$

33e

To reduce the influence of discretization error on LWS change attribution results, the average values of the elasticity coefficients in the pre-change (subscript pre) and post-change (subscript post) periods were used in practice (Jiang et al. 2015; Yang et al. 2020). Therefore, the relative contribution to change in LWS by arbitrary variable x₄ can be expressed as follows:

${\omega _{{x_4}}}=\frac{{\Delta S_{l}^{{{x_4}}}}}{{\Delta {S_l}}}=\frac{1}{2}\left( {{\eta _{{S_{{x_4},pre}}}}+{\eta _{{S_{ls,post}}}}} \right)\frac{{\Delta {x_4}}}{{{x_4}}}\frac{{{S_l}}}{{\Delta {S_l}}} \times 100\%$ , x₄ = S_ls, W_n, W_a, W_s, S_e, α₁ (34)

in which ${\omega _{{x_4}}}$ is the contribution of change in arbitrary variable x₄ (x₄ = S_ls, W_n, W_a, W_s, S_e and α₁), ΔS_l is the change in LWS from the pre-change period to the post-change period, $\Delta S_{l}^{{{x_4}}}$is the change in LWS induced by arbitrary variable x₄, Δx₄ is the change in arbitrary variable x₄ from the pre-change period to the post-change period.

4.1 Estimation of LWS change in Lake Dianchi

4.1.1 Parameter calibration

Figure 2 shows that, for the upper water level limit, two breakpoints occurred, in 1988 and in 2012. Thus, the 56-year study period was split into three sub-periods, namely sub-period I under Plan-1956 (1964 to 1988), sub-period II under Plan-1988 (1989 to 2012), and sub-period III under Plan-2012 (2013 to 2019). Because different regulation plans were executed in the three sub- periods, parameter α₁ was calibrated for each sub-period. Previous studies have shown that, for a specific basin, the DWBM parameters (α_2,α_3,S_b*, and k) mainly depend on land use/cover (Xu et al. 2014; Gao et al. 2016). Because land use/cover change was more significant in the Lake Dianchi basin during recent decades owing to rapid urbanization which was characterized by an increase in constructed land and a decrease in grassland and cultivated land (Zhao et al. 2012; Wang et al. 2021), the four DWBM parameters were individually calibrated4.1.2 Model evaluation for the sub-periods involved in this study. To test the proposed model, the observed data in each sub- period were further divided into two groups: the first (approximately 70% of the data) was used for calibration, and the second was used for validation.

In the calibration procedure, when the proposed model matched the measured data reasonably well, the parameter α₁ was estimated as 1.64 for sub-period I, and the values of the DWBM parameters were α₂ = 4.55, α₃ = 4.35, S_b*=130 mm, and k = 0.50. A comparison of observed LWS with the estimated results from the proposed model, shown in Fig. 6, gave an RRMSE of 2.52%, and MAPE of 2.22%. For sub-period II, the values of model parameters for the fitted curve were α₁ = 2.22, α₂ = 4.55, α₃ = 3.57, S_b*=100 mm, and k = 0.45, with a RRMSE of 5.47%, and MAPE of 3.97%. For sub-period III, the values of model parameters are α₁ = 2.63, α₂ = 4.55, α₃ = 3.57, S_b*=100mm, and k = 0.45, with a RRMSE value of 2.17%, and MAPE value of 1.63%. For the study period as a whole, these resulted in an RRMSE of 4.08%, and MAPE of 2.88%.

The statistical indicators for model evaluation are shown in Table 3. The three studied sub-periods had RRMSE values of less than 10%, falling into the criteria of excellent performance. This indicated that the proposed model was accurately calibrated for annual LWS change estimation in Lake Dianchi. Comparisons of estimated results with observed LWS in the three calibration periods are shown in Fig. 6.

Table 3

Performance metrics of annual LWS estimates from the proposed model
Period		MAPE (%)	RRMSE (%)	CR (%)	B (10⁶ m³)	D (10⁶ m³)
Sub-period I	Calibration	2.22	2.52	82	119	42
Sub-period I	Validation	4.00	5.07	38	86	76
Sub-period II	Calibration	3.97	5.47	38	116	75
Sub-period II	Validation	3.46	4.81	50	133	65
Sub-period III	Calibration	1.63	2.17	100	159	31
Sub-period III	Validation	1.18	1.42	100	172	14
Note: CR is the containing ratio, B is the average bandwidth, and D is the average deviation amplitude.

4.1.2 Model evaluation

The validation sets were used to evaluate the performance of the calibrated model. Figure 6 shows comparisons of estimated and measured LWS for the three sub-periods with low RRMSEs, as listed in Table 3, which all meet the criteria of excellent performance. Table 3 also presents three uncertainty indicators for 95% confidence limits of total uncertainty for LWS during the three sub-periods. Overall, the results indicate that the proposed model has the ability to reasonably replicate LWS dynamics in Lake Dianchi.

4.2 Quantitative attribution of LWS change in Lake Dianchi

4.2.1 Changes in the supply term, demand term and parameter α₁

The proposed model with an annual time step was used to analyze attribution in this study. For convenience, the mean annual values of the arbitrary variable x₄ (x₄ = S_ls, W_n, W_a, W_s, S_e, and α₁) during the three sub-periods were denoted as $x_{4}^{{\text{I}}}$, $x_{4}^{{{\text{II}}}}$, and $x_{4}^{{{\text{III}}}}$, respectively. The change in the arbitrary variable x₄ was defined as

${\Delta _{{\text{I}} \to {\text{II}}}}\left( {{x_4}} \right)=x_{4}^{{{\text{II}}}} - x_{4}^{{\text{I}}}$ , ${\Delta _{{\text{II}} \to {\text{III}}}}\left( {{x_4}} \right)=x_{4}^{{{\text{III}}}} - x_{4}^{{{\text{II}}}}$ (35)

in which ${\Delta _{{\text{I}} \to {\text{II}}}}\left( {{x_4}} \right)$ is the change of x₄ from sub-period I to sub-period II, and ${\Delta _{{\text{I}} \to {\text{II}}}}\left( {{x_4}} \right)$ is the change of x₄ from sub-period II to sub-period III.

As shown in Table 4, in sub-period I W_{0, I} was equal to 776 million m³, of which approximately 62% was attributed to $W_{n}^{{\text{I}}}$ through the natural hydrological cycle. In sub-period II, $W_{0}^{{{\text{II}}}}$ increased to 966 million m³ (i.e., ${\Delta _{{\text{I}} \to {\text{II}}}}\left( {{W_0}} \right)=$190 million m³). Because the stored water resources increased by 145 million m³, the portion of $W_{n}^{{{\text{II}}}}$ decreased to 54%. In sub-period III, because channel inflow of 397 million m³ was transferred into the lake through the NDWTP, the total usable water increased to 1506 million m³ (i.e., ${\Delta _{{\text{II}} \to {\text{III}}}}\left( {{W_0}} \right)$= 54 million m³), and the proportion of $W_{n}^{{{\text{III}}}}$ sharply decreased to 39%. These results demonstrate that the composition of the total usable water resources in Lake Dianchi changed from being mainly dependent on the natural hydrological cycle in sub-periods I and II to being controlled by human activities in sub-period III. Meanwhile, owing to the increase in the maximum controllable level, the effective storage capacity increased from 510 million m³ in sub-period I to 571 million m³ in sub-period II, which indicates a S_e increase of 61 million m³ in sub-period II relative to sub-period I. Similarly, a S_e increase of 31 million m³ occurred in sub-period III relative to sub-period II. The values of parameter α₁ for the three sub-periods were 1.64, 2.22 and 2.63, which indicated a ${\Delta _{{\text{I}} \to {\text{II}}}}\left( {{\alpha _1}} \right)$ value of 0.58 and a ${\Delta _{{\text{II}} \to {\text{III}}}}\left( {{\alpha _1}} \right)$ value of 0.41.

Table 4

Mean and relative changes of main factors attributable to LWS changes in Lake Dianchi
Sub-period	Supply						Demand		Parameter
	W_n (10⁶ m³)		W_a (10⁶ m³)		W_s (10⁶ m³)		S_e (10⁶ m³)		α₁
	Mean	Change	Average	Change	Mean	Change	Mean	Change	Mean	Change
I	478	–	0	–	298	–	510	–	1.64	–
II	523	45	0	0	442	144	571	61	2.22	0.58
III	579	56	397	397	519	77	602	31	2.63	0.41

4.2.2 Quantitative attribution of LWS change

Overall, elasticity coefficients of LWS were positively correlated with W_n, W_a, W_s, S_ls, S_e and parameter α₁ during the three sub-periods (Table 5). The values were largest for S_ls, intermediate for S_e and α₁, and smallest for W_a. Over time, the value of η_Se increased during the three sub-periods but both η_Wn and η_α1 decreased, which suggests that LWS became more sensitive to the lake water level regulations but less sensitive to climate change and lake storage efficiency. In sub-period III with the inter-basin water transfer through the NDWTP, the value of η_Wa was 0.02, indicating that a 10% increase in channel inflow from outside the basin would increase LWS by 0.2%.

Table 5

Quantitative contributions of LWS change in Lake Dianchi
Sub-period	Elasticity of LWS						Quantitative contribution to LWS change (%)
Sub-period	η_Sls	η_Wn	η_Wa	η_Ws	η_Se	η_α1	ω_Sls	ω_Wn	ω_Wa	ω_Ws	ω_Se	ω_α1
I	0.77	0.06	0	0.04	0.13	0.31	–	–	–	–	–	–
II	0.69	0.07	0	0.06	0.19	0.20	0.0	5.3	0.0	14.9	15.9	63.7
III	0.64	0.02	0.01	0.02	0.30	0.12	0.0	7.8	12.9	10.2	23.2	45.9
Average	0.70	0.05	0.01	0.04	0.20	0.21	0.0	6.6	6.5	12.6	19.6	54.9

The relative contributions of variations in W_n, W_a, W_s, S_ls, S_e and parameter α₁ are also shown in Table 5. The main factors influencing the change in LWS varied in the different sub-periods. The increased LWS in sub-period II was controlled by changes in lake storage efficiency (up to 63.7%), whereas the change in W_n had a minor impact on LWS increase. For sub-period III, the contribution of changes in W_a and S_e to the increased LWS was 12.9% and 23.2%, respectively (Table 5), whereas the contribution of changes in W_n further reduced to 7.8%. Notably, the changes in W_a and S_e owing to the alteration of lake regulation strategies became more pronounced.

4.3 Controls of LWS dynamics in regulated Lake Dianchi

Figure 7 depicts the scatter plots of the actual storage ratio S_la/W₀ against S_e/W₀ in Lake Dianchi during the three sub-periods. Similar to the observed behavior in the original Budyko curves, as S_e/W₀ increased, S_la/W₀ increased within the two limits. Clearly, the effect of lake storage efficiency α₁ on S_la/W₀ was minimal under the two extreme conditions. This was because under these conditions, LWS was dominated by W₀ and S_e, respectively. When S_e/W₀ was approximately 1, the actual storage ratio S_la/W₀ showed maximum sensitivity to α₁. One possible explanation is that in this condition, both W₀ and S_e had the same control over active LWS change, and thus the lake regulation rules imposed a larger impact on LWS change. The results indicate that the LWS dynamics in this studied lake are subject to the supply and demand of the lake water circulation system. Most of the data points fall within the water-storage-limited zone, i.e., S_e/W₀ < 1, as shown in Fig. 7, which indicates that Lake Dianchi is an apparent water-storage-limited lake.

<Fig. 7>

It should be noted that during each sub-period under the same regulation scheme, the α₁ value can be fixed as a constant, which enables the model to predict further LWS change in regulated lakes. According to a literature survey, the Lvfeng Gate at the outlet river has an insufficient discharge capacity. To minimize the flood risk, the regulators limited the water level of Lake Dianchi during sub-period I below 1886.7 m, which was lower than the assigned value of 1887.2 m in Plan-1964. If the actual upper limit of water level was used for calculation, the effective storage capacity S_e in sub-period I was reduced to 358 million m³. As a result, the calibrated value of α₁ increased to 2.00 for the best fitted curve. This indicates that the value of α₁ is closely related to S_e. The smaller the S_e value, the larger the α₁ value.

4.4 Comparison with the water balance method

On the basis of the reservoir routing model proposed by Hanasaki et al. (2006) (thereafter referred to as HM), the water balance method was used to assess the relative performance of the proposed model. The non-irrigation HM was selected to estimate outflow, which is expressed as (Hanasaki et al. 2006):

$$Q_{0}^{t}=\left\{ \begin{gathered} \frac{1}{C}\frac{{S_{l}^{{t - 1}}}}{{{S_{l*}}}}Q_{{in}}^{t}, {\text{IR}} \geqslant 0.5 \hfill \\ \frac{1}{C}\frac{{S_{l}^{{t - 1}}}}{{{S_{l*}}}}{\left( {\frac{{{\text{IR}}}}{{0.5}}} \right)^2}Q_{{in}}^{t}+\left[ {1 - {{\left( {\frac{{{\text{IR}}}}{{0.5}}} \right)}^2}} \right]Q_{{in}}^{t}{\text{,}} {\text{IR}}<0.5 \hfill \\ \end{gathered} \right.$$

36

in which ${\text{IR}}\left( {={{{S_{l*}}} \mathord{\left/ {\vphantom {{{S_{l*}}} {{I_0}}}} \right. \kern-0pt} {{I_0}}}} \right)$ is the impoundment ratio of the lake, where I₀ is the mean annual inflow (m³/yr); Q_in is the inflow; and C is a dimensionless coefficient, which was fixed at 0.85 by Hanasaki et al. (2006). In the present study, the value of C was calibrated within the range of 0.1 to 1.5.

The annual LWS estimated by the proposed model and the HM-based water balance method is plotted in Fig. 8. It can be seen that the proposed model matches quite well with the observed LWS dynamics, while HM-based water balance method is likely to underestimate LWS, especially in dry years (e.g., 2009–2013). The proposed model has a lower RRMSE of 4.08% than that of HM-based water balance method of 16.37%. These results occurred most likely because HM ignores the impact of the minimum drawdown level on outflow, which may produce a large error in outflow, and eventually in the closure term of the water balance (i.e., LWS change). By comparison, the proposed model not only incorporates the impacts of both total usable water resources and water level control within Budyko’s demand-supply framework, but also eliminates the outflow error from the estimated LWS change. Thus, it may be a better candidate for estimating LWS change in regulated lakes.

4.5 Implications of the proposed model

Adaptive management of water resources in regulated lakes is the key challenge for lake managers and policymakers. The present study provides a simple method for estimating the variation of LWS in regulated lakes from the perspective of the water supply and demand balance. On this basis, the elasticity method can be applied to separate the impacts of the main controlling factors such as lake regulation activities and inter-basin water transfer on LWS change in regulated lakes. Owing to its simplicity and low data requirements, this method can be feasibly implemented to determine low-cost but effective water resources management strategies. This model can be applicable for regulated lakes to predict future LWS under the specified regulation scheme, reconstruct outflow time series, and assess the influence of inter-basin water transfer on lake level fluctuations.

Following Budyko’s supply–demand framework, this study proposed a water balance model to estimate annual LWS change without the need for water output information. The study found that the proportion of active LWS was controlled by the total usable water resources and the effective storage capacity of the lake. The impacts of the main driving factors on LWS were investigated using the elasticity method. The proposed model was applied to Lake Dianchi, a typical regulated lake located in southwest China. The main conclusions derived from the present study are as follows.

(1) Plotting the actual storage ratio S_la/W₀ against the potential storage ratio S_e/W₀ in Lake Dianchi showed a Budyko-type relationship, exhibiting a monotonically increasing shape within two limits. The results revealed that total usable water resources and effective storage capacity controlled the variability of LWS change in the regulated lake.

(2) The proposed model was capable of estimating LWS change in a regulated lake with reasonable accuracy. Compared with the HM-based water balance method, the model was more effective at capturing LWS dynamics in Lake Dianchi because it accounted for the effects of total usable water resources, water level control and lake regulation with the estimation error of outflow eliminated in the water balance.

(3) The lake regulation strategy was the most important factor affecting the change in LWS in Lake Dianchi, contributing to an increase of 54.9% in LWS on average, while the inter-basin water transfer made a relatively limited contribution in determining LWS change. This is especially useful for adaptive management of water resources in the regulated lake.

Author Contributions M Zhou: Conceptualization, Methodology, Software, Formal analysis, Writing - Original Draft. L Xiong: Conceptualization, Resources, Writing - Review & Editing, Project administration, Funding acquisition. G Chen: Data collection, Writing-Review & Editing. J Chen: Resources, Writing - Review & Editing. DD Liu: Writing - Review & Editing. S Li: Software, Writing - Review & Editing. All authors read and approved the final manuscript.

Funding This research was financially supported by National Natural Science Foundation of China (NSFC Grants 41890822 and 51525902), the National Key R&D Program of China (2017YFC0405901), and the Ministry of Education “Plan 111” Fund of China (B18037), all of which are greatly appreciated.

Availability of Data All data are available from the corresponding author upon reasonable request.

Ethical Approval The author promises to compliance with Ethical Standards.

Consent to Participate Informed consent was obtained from all individual participants included in the study.

Consent to Publish The participants have consented to the submission of the research article to the journal.

Competing Interests The authors declare they have no conflict of interest.

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Highlights.docx

Estimation and attribution of water storage changes in regulated lakes based on Budyko's supply–demand framework

Status:

Journal Publication

Version 1

Abstract

Figures

1 Introduction

2 Study Area And Data Sources

2.1 Study area

2.2 Data sources

3 Methods

3.1 Budyko-based LWS change estimation model

3.1.1 Lake water balance equation

3.1.2 Rearrangement of the lake water balance equation

3.1.3 Budyko-based model for LWS change estimation

3.1.4 Estimation of basin runoff

3.2 Parameter calibration and uncertainty analysis

3.3 Attribution analysis of change in LWS

4 Results And Discussion

4.1 Estimation of LWS change in Lake Dianchi

4.1.1 Parameter calibration

4.1.2 Model evaluation

4.2 Quantitative attribution of LWS change in Lake Dianchi

4.2.1 Changes in the supply term, demand term and parameter α₁

4.2.2 Quantitative attribution of LWS change

4.3 Controls of LWS dynamics in regulated Lake Dianchi

4.4 Comparison with the water balance method

4.5 Implications of the proposed model

5 Conclusions

Declarations

References

Supplementary Files

Status:

Journal Publication

Version 1

Purpose	Statistical metrics	Equation	Perfect value
Model performance evaluation	Relative root mean square error (RRMSE)	\({\text{RRMSE}}=\frac{1}{{{\mu _m}}}\sqrt {\frac{1}{T}\sum\limits_{{t=1}}^{T} {{{\left( {v_{m}^{t} - v_{e}^{t}} \right)}^2}} }\)	0
Model performance evaluation	Mean absolute percentage error (MAPE)	\({\text{MAPE}}=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left\| {\frac{{v_{e}^{t} - v_{m}^{t}}}{{v_{m}^{t}}}} \right\|}\)	0
Uncertainty analysis	Containing Ratio (CR)	\({\text{CR}}=\frac{N}{T}\)	1
	Average Bandwidth (B)	\(B=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left( {v_{{95\% \hbox{max} }}^{t} - v_{{95\% \hbox{min} }}^{t}} \right)}\)	0
	Average Deviation Amplitude (D)	\(D=\frac{1}{T}\sum\limits_{{t=1}}^{T} {\left\| {v_{e}^{t} - \frac{1}{2}\left( {v_{{95\% \hbox{max} }}^{t}+v_{{95\% \hbox{min} }}^{t}} \right)} \right\|}\)	0
Note. \({v_m}\), \({v_e}\)are measured and estimated values, respectively; \({\mu _m}\)is the average value of the measured data; \(V_{{95\% \hbox{max} }}^{t}\), \(V_{{95\% \hbox{min} }}^{t}\) represent the upper and lower bound values of the 95% confidence level, respectively; T is the length of data series, and N is the number of observed value that fall within the 95% confidence limits.

Estimation and attribution of water storage changes in regulated lakes based on Budyko's supply–demand framework

Status:

Journal Publication

Version 1

Abstract

Figures

1 Introduction

2 Study Area And Data Sources

2.1 Study area

2.2 Data sources

3 Methods

3.1 Budyko-based LWS change estimation model

3.1.1 Lake water balance equation

3.1.2 Rearrangement of the lake water balance equation

3.1.3 Budyko-based model for LWS change estimation

3.1.4 Estimation of basin runoff

3.2 Parameter calibration and uncertainty analysis

3.3 Attribution analysis of change in LWS

4 Results And Discussion

4.1 Estimation of LWS change in Lake Dianchi

4.1.1 Parameter calibration

4.1.2 Model evaluation

4.2 Quantitative attribution of LWS change in Lake Dianchi

4.2.1 Changes in the supply term, demand term and parameter α1

4.2.2 Quantitative attribution of LWS change

4.3 Controls of LWS dynamics in regulated Lake Dianchi

4.4 Comparison with the water balance method

4.5 Implications of the proposed model

5 Conclusions

Declarations

References

Supplementary Files

Status:

Journal Publication

Version 1

4.2.1 Changes in the supply term, demand term and parameter α₁