Impacts of Climate Change on Mean Annual Water Balance for Watersheds in Michigan, Usa

Evaluation of water balance at the watershed scale is a fundamental step for estimating streamflow in watersheds. Mean annual water balance of 17 watersheds across Michigan were evaluated by comparing observed streamflow with simulated streamflow estimated using Fu's Equation, which is based on the Budyko Hypothesis. The Budyko Hypothesis describes mean annual water balance as a function of available water and energy. Impact of long-term climatic controls (e.g., precipitation, potential evapotranspiration (ETP)) on mean annual water balance was also investigated with Fu's Equation. Results indicated that observed streamflow ranged from 237 to 529 mm per year, with an average of 363 mm per year in the study watersheds during 1967–2011. On average, 40% of long-term precipitation in the study watersheds was converted into surface runoff. The performance of Fu's Equation in estimating mean annual streamflow resulted in Root Mean Square Error (RMSE) value of 64.1 mm/year. Mean annual streamflow was sensitive to changes in mean annual precipitation, and less sensitive to changes in mean annual ETp in the watersheds. With the increase of baseflow index (BFI), mean annual streamflow was less sensitive to climate change. Overall, different contributions of baseflow to 3566 streamflow modified the impact of climate controls on mean annual water balance in the baseflow-dominated watersheds.


Introduction
Water balance refers to the quantitative description of the hydrologic cycle.It is often expressed with water balance equations (i.e., the relationship between input and output of water through an area during a given time period).A water balance model is generally selected based on specific project requirements, available data, and the application scope of the model.Empirical data of catchment all over the world indicate that the long-term water balance is primarily controlled by water supply (i.e., precipitation) and energy demand (i.e., potential evapotranspiration), these relatively simple models have been developed [1][2][3][4].The Budyko Hypothesis, the relationship between actual evapotranspiration ratio (i.e., the ratio of actual evapotranspiration to precipitation) and climate aridity index (i.e., the ratio of potential evapotranspiration to precipitation), is a useful tool to examine watershed water balance and to investigate the effects of climate change and watershed characteristics on the hydrologic cycle [5][6][7][8][9].
Although a variety of equations satisfying the Budyko Hypothesis can be found in previous studies [4,[10][11][12][13][14][15], Budyko-type equations with optimizable parameters are particularly useful for modeling mean annual and annual water balance in individual watersheds [16,17].Fu's Equation, an analytical solution of the Budyko Hypothesis, is a one-parameter conceptual model derived from dimensional analysis and mathematical reasoning (rather than a simple empirical fit to hydrologic data) [12].Fu's Equation can provide reasonable results for estimating mean annual and annual water balances in most individual watersheds [15,18].Based on its simplicity, Fu's Equation was utilized in this study to evaluate long-term water balance at catchment scales.
Long-term water balance is affected by climatic controls (i.e., precipitation (P) and potential evapotranspiration (ETp)) according to the Budyko Hypothesis.The Budyko water balance approach for estimating the climatic sensitivity has been widely applied in previous studies [8,[19][20][21][22].However, studies relating groundwater (e.g., baseflow) to Budyko Hypothesis were relatively rare.Wang et al. [23] explored the effects of soil texture and groundwater (e.g., baseflow) on mean annual and annual water balances of watersheds in the Nebraska Sand Hills, based on the Budyko Hypothesis.Results suggested that soil texture may greatly modify the influence of climate on regional water balance; and a water storage term needed to be included in the Budyko Hypothesis on annual time scales when baseflow contribution was significant.
This study is a continuous work based on [24], in which 17 watersheds have been used to develop regression equations for estimating baseflow and baseflow index.On average, baseflow accounted for 70% of the total streamflow in those 17 watersheds [24].The specific objective of this study were to (1) test the applicability of Fu's Equation in estimating mean annual water balances for 17 watersheds in Michigan; and (2) investigate the relationship between baseflow and climatic sensitivity in Michigan watersheds using Fu's Equation.The evaluation of watershed water balance in this study was intended to provide quantitative insight into water balance response to climatic factors in various watersheds, especially in baseflow-dominated watersheds.

Data Used
Watershed characteristics and meteorological variables used for water balance calculations in this study include: Watershed location (latitude of the gaging stations), annual precipitation (P), annual streamflow (Q), maximum daily temperature (averaged over all days in the month) (Tmax) and minimum daily temperature (averaged over all days in the month) (Tmin), and solar radiation (Ra).Annual precipitation and monthly average maximum and minimum daily temperature for the study watersheds were derived from Parameter-elevation Regressions on Independent Slopes Model (PRISM) climate analysis system [25].All datasets mentioned above were compiled using ArcGIS 10 [26] for a period of 1967 to 2011.Streamflow data were obtained from USGS (United States Geological Survey) gaging stations [27].The Eckhardt filter method [28] in Web-based Hydrograph Analysis Tool program was used to partition daily streamflow records into direct runoff and baseflow for the study period of 1967 to 2011 [29].The default recess constant and BFImax values of 0.98 and 0.8 were employed, respectively.The specific principle of the recursive partitioning algorithm was shown in detail in the study by Zhang et al. [24].Then, baseflow index (BFI) was calculated by dividing mean annual baseflow by mean annual streamflow to quantify groundwater contributions to streamflow in the 17 study watersheds.

Potential Evapotranspiration Calculation
The Hargreaves method [30] is based on time series of average maximum and minimum temperature data and solar radiation.It has been widely used to estimate ETP in previous studies [31][32][33].The equation can be expressed as [30]: where ETp is the potential evapotranspiration, mm/d; Tmax and Tmin are the maximum monthly temperature and minimum monthly temperature, respectively, °C; Ra is the extraterrestrial solar radiation, mm/d.Daily solar radiation of individual gaging stations was calculated based on the latitude values (shown in Table 1) using the calculator of extraterrestrial solar radiation tool [34].Monthly time series of average maximum and minimum temperature data along with the computed solar radiation were used to calculate monthly ETp in mm/d using Equation (1).Annual ETp in the study watersheds was derived from summing monthly ETp of each year.

Water Balance Modeling Based on Budyko Hypothesis
Budyko [3,4] reported that mean annual ETa is primarily controlled by available water and energy.Budyko Hypothesis is a model that represents ETa/P ratio as a function of climatic aridity index.It can be expressed as: where ETa/P is the ETa ratio; ETp/P is the climate aridity index; F is an empirical function.
The relationship presented in Equation ( 2 To incorporate the effects of factors such as vegetation, soil water storage, and rainfall seasonality, among others, on water balance, Budyko-type equations with adjustable parameters have been developed for various applications [5,[12][13][14]23].Fu's Equation contains a parameter (w, w > 1) that represents the combined effects of climatic conditions (e.g., rainfall seasonality) and catchment characteristics (e.g., vegetation cover, soil properties and catchment topography) on the water balance [15,35].Fu's Equation can be expressed as: In this study, ETa was derived from water balance equation (i.e., ETa = P − Q) and w was fitted using SOLVER in Microsoft Excel 2010.The average fitted w (w = 1.95) was utilized for calculating mean annual Q using the following equation: RMSE (Root Mean Square Error) was used to evaluate the performance of Fu's Equation (w = 1.95) in the estimation of mean annual streamflow of 17 watersheds in Michigan.The corresponding equation was expressed as: where Qobs,i is the observed mean annual streamflow (mm/year); Qpred,i is the predicted mean annual streamflow (mm/year); n is the total number of the study watersheds (n = 17).

Climate Sensitivity of Streamflow Based on the Budyko Hypothesis
The sensitivity of mean annual Q to variations in mean annual P and ETp in individual watersheds was evaluated using an analytical framework proposed by Roderick and Farquhar [8].The framework described that changes in Q in a watershed is a function of changes in climate variables (i.e., P and ETp) and watershed properties (w) (i.e., changes to climate variability, topography, soil type and vegetation, etc.).Following Roderick and Farquhar [8] and neglecting variation in watershed properties, the influence of mean annual P and ETp on variation in mean annual Q can be expressed as [8,36]: where partial derivatives (i.e., ∂Q/∂P and ∂Q/∂ ETP) represent Q sensitivity to a 1-unit variation in mean annual P and ETp.Larger ∂Q/∂P and ∂Q/∂ETP values represent greater influence of variation in mean annual P and ETp on Q.
In the Equation ( 7), ∂Q/∂P and ∂Q/∂ETp were computed from Fu's Equation (Equation ( 5)).The mathematical expressions for computing ∂Q/∂P and ∂Q/∂ ETP can be expressed as [8]: Rearranging Equation ( 7) by dividing by mean annual Q, the percent change in mean annual Q response to percent change in mean annual P and ETP was computed as [8]: where the functional expressions in brackets are the sensitivity coefficients of Q responses to P and ETP changes, respectively.Sensitivity coefficients in Equation ( 10) refer to the proportional change in Q relative to a 1% change in P and ETP.For example, sensitivity coefficients of Q responses to P and ETP changes in Equation (10) were 0.5 and −0.5, respectively, indicating a 10% increase in P increased Q by 5% while a 10% increase in ETP decreased Q by 5%.

Mean Annual Water Balance in the Study Watersheds
The spatial variation in ETp/P in the study watersheds was ranged from 0.94 to 1.17 (Table 1).This indicates that all the study watersheds are in a sub-humid climatic zone according to a study by Ponce et al. [37], in which the authors reported that climatic spectrum could be divided into eight types depending on ETp/P ranges.Xu et al. [38] also reported similar ETp/P ratios, which vary from 0.87 to 1.33 in 55 watersheds across the Midwest United States.Mean annual ETa/P ratio and runoff coefficient (Q/P) in the study watersheds varied from 0.34 to 0.71 and from 0.29 to 0.66, respectively (Table 1).Low ETa/P ratio under similar P was likely attributed to snowiness as Q in the Trap Rock River watershed mainly originated from snow fall and spring snowmelt [24].Berghuijs et al. [39] also reported that snowy catchments have a high runoff ratio in context of the Budyko hypothesis.On average, 40% of long-term P in the study watersheds was converted into surface runoff.Mean annual Q/P ratios were large for watersheds with high BFI values, while the corresponding ETa/P ratios were low.As shown in Figure 1, the relationship between mean annual ETa/P and ETp/P ratios in the study watersheds satisfied Fu's curve with w = 1.95.Results indicated that estimated mean annual Q using Fu's Equation (Equation ( 5)) agreed relatively well with observed mean annual Q with RMSE value of 64.1 mm/year (Figure 2).

Impact of Climatic Controls on Mean Annual Water Balance
Q sensitivity analysis indicated that mean annual Q increased from 12.6% in the Trap Rock River watershed to 20.6% in the Stony Creek watershed with an average value of 16.7% when mean annual P increased by 10% (Table 2).A 10% increase in ETp decreased Q from 2.7% in the Trap Rock River watershed to 10.9% in the Stony Creek watershed with an average value of 6.9% (Table 2).Results also showed that streamflow sensitivity to changes in P and ETp had a decreasing trend from north to south.Overall, mean annual Q was sensitive to variations in mean annual P and less sensitive to variations in mean annual ETp for the period of 1967-2011 in all 17 watersheds.Similar results were found in a study conducted by Donohue et al. [40], in which the authors reported that Q increased by 7 mm/year with a 10 mm/year increase in P, and decreased by 4 mm/year for the same increase in ETp in Australia for the period of 1981-2006.Roderick and Farquhar [8] applied this method in the semi-arid Murray Darling Basin in Australia and indicated that a 10% change in long-term average P yielded approximately 26% change in average Q.Herein, streamflow deviation ratio (SDR) (i.e., the ratio of the standard deviation of annual Q to that of annual P) proposed by Koster and Suarez [41] was used in this study to demonstrate the sensitivity of the variability in inter-annual Q to the variation in inter-annual P. Results indicated that SDR ranged from 0.35 in the Manistee River watershed to 1.05 in the Macatawa River watershed, suggesting that the majority of inter-annual P variability became inter-annual Q variability for the Macatawa River watershed, while inter-annual Q variability was largely less sensitive to variability in inter-annual P for the Manistee River watershed.SDR was low for watersheds with high BFI values, indicating that inter-annual Q variability was largely less sensitive to variability in inter-annual P for watersheds with high BFI values.It seemed that different contributions of baseflow to streamflow modified the impact of climate controls on water balance in the baseflow-dominated watersheds.That is to say, mean annual Q was less sensitive to climate change with the increase of BFI.Zeng and Cai [42] attributed ET variance to both the mean and variance of climatic variables by extending the framework of Koster and Suarez [41].Results showed that catchment storage change played a significant role to buffer the inter-and intra-annual variance of ET in the Murray-Darling Basin.

Compared with Other Similar Studies
All 17 study watersheds were located in a sub-humid climatic zone and thus climatic conditions of these watersheds were similar.However, ETa/P ratios varied across the study watersheds.Previous studies suggested that differences in ETa/P ratios under the same climatic aridity index were explained by land cover and/or soil texture [13,14,23].Watersheds, such as Stony Creek watershed and Clinton River watershed had similar climate conditions (P and ETp/P), with ETa/P ratios of 0.71 and 0.59, respectively (Table 1).This may be due to the different percentages of soil types and land cover [43,44], with hydrologic soil group B comprising 73% and 57% of Stony Creek watershed and Clinton River watershed, respectively.Stony Creek watershed was mainly covered by forest (37%) while developed land cover constituted 65% of the Clinton River watershed.Overall, the varying ETa/P ratios in the study watersheds could be the result of combinatory effects of land cover and soil properties.The average Q/P value of 0.42 during the 1967-2011 study period suggested that about 40% of long-term P in the study watersheds was converted into surface runoff.Similar results were found by Tekleab et al. [17], where Q/P ratios varied from 0.21 to 0.70 for 20 watersheds in the Upper Blue Nile basin.By contrast, Q/P ratios of 34 subbasins in the Nebraska Sand Hills reported by Wang et al. [23] were very low, ranging between 0.01 and 0.18, which can be explained by the high infiltration capacity of sandy soils in the Nebraska Sand Hills.
On average, 70% streamflow was contributed by baseflow in the studied watersheds [24].As shown in Figure 3, the correlation coefficient between Q/P and BFI in the remaining 15 watersheds (Trap Rock River and Macatawa River watersheds were excluded) approaches to 0.5.It is suggested that Q/P ratios increased with the increase of BFIs.Similar results were reported by [23], which indicated that higher BFI values and Q/P ratio could be explained by higher groundwater recharge.Watersheds with similar BFI values would have similar ETa/P ratios (e.g., Thornapple River and Battle Creek) (Table 1).Inversely, if BFI values were different, ETa/P ratios would be different.Although average annual water balance was principally controlled by available water and energy (i.e., P and ETp), factors such as rainfall seasonality, root zone storage capacity and snowiness have also been shown to be major controls on long-term water balance behavior [13,45].Since Q/P ratio for the Trap Rock River and Macatawa River watersheds seemed unusual compared to the other study watersheds; they were not used for the analysis.As mentioned in Section 3.1.,large Q in the Trap Rock River watershed could be the influence of heavy snow and large amounts of spring snowmelt.Low BFI values in the Macatawa River watershed could be explained by large proportions of agricultural and urban land uses as well as soil texture (dominated by hydrologic soil group C) that would reduce the rate of water transmission of the underlying aquifer and groundwater discharge into the streams [24].Similar findings were reported in previous studies [8,23].Wang et al. [23] indicated that soil texture altered the influence of climate on regional water balances to large extent and water storage should be included in the Budyko Hypothesis for baseflow-dominated watersheds.

Limitations
ETp is an important variable for estimating ETa and climate aridity index in hydrological modeling.Limited to the data availability, the Hargreaves method [30] was selected to calculate ETp in this study.Since the Hargreaves equation was originally calibrated using data from California, the transferability of this equation to other regions is quite limited.The reliability of ETp estimates can be improved by adding more relevant input variables [46,47].Thus, the impact of ETp calculated by Hargreaves method on mean annual water balance at watershed scales need to be further explored.Although Fu's Equation was employed to estimate the role of climate changes on changing water balance conditions in this study, comparison of other empirical equations based on Budyko Hypothesis is necessary to be conducted in the future studies.In addition, watershed characteristics such as vegetation cover, soil properties and watershed topography were integrated in parameter w in Fu's equation [15]; however, values of w in individual watersheds in this study were simply fitted between the observed ETa/P and ETp/P ratios.This may result in limitations in using Fu's Equation for mean annual water balance estimation.Some studies have focused on the development of w estimation to improve the predictive ability of Fu's Equation [45,[48][49][50].In addition, watershed boundaries were not considered in this study.This may result in water exchange within the adjacent watersheds, thereby limiting the accuracy of Fu's Equation in estimating mean annual water balance in the study watersheds.

Conclusions
Mean annual water balances in 17 watersheds across Michigan were estimated using Fu's equation, which is based on the Budyko Hypothesis.The role of climate controls (e.g., P and ETp) on mean annual water balance was investigated with Fu's Equation.Results indicated that estimated mean annual Q using Fu's Equation agreed relatively well with observed mean annual Q with RMSE value of 64.1 mm/year.On average, 40% of long-term P in the study watersheds was converted into surface runoff.Mean annual Q/P ratios were large for watersheds with high BFI values, while the corresponding ETa/P ratios were low, suggesting that Q was closely related to regional groundwater discharge.Climate sensitivity of mean annual Q showed that a 10% increase in mean annual P increased mean annual Q by 16.7%, while a 10% increase in mean annual ETp decreased Q by 6.9% on average.This suggested that mean annual Q was sensitive to changes in mean annual P and less sensitive to changes in mean annual ETp in all 17 watersheds.It seemed that different contributions of baseflow to streamflow modified the impact of climate controls on annual water balance in the baseflow-dominated watersheds.

Figure 1 .
Figure 1.Comparison of observed and calculated (using Fu's Equation) ETa/P in the study watersheds.

Figure 2 .
Figure 2. Predicted vs. observed mean annual streamflow in the study watersheds.

Table 1 .
Gaging stations and water balance components of seventeen watersheds in Michigan.

Table 2 .
Sensitivity of mean annual Q to variations in mean annual P and ETp in the study watersheds in Michigan.