Department of Civil Environmental and Construction Engineering, University of Central Florida, Orlando, Florida, USA
Corresponding author: D. Wang, Department of Civil, Environmental, and Construction Engineering, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816, USA (email@example.com)
 Long-term climate is the first-order control on mean annual water balance, and vegetation and the interactions between climate seasonality and soil water storage change have also been found to play important roles. The purpose of this paper is to extend the Budyko hypothesis to the seasonal scale and to develop a model for interannual variability of seasonal evaporation and storage change. A seasonal aridity index is defined as the ratio of potential evaporation to effective precipitation, where effective precipitation is the difference between rainfall and storage change. Correspondingly, evaporation ratio is defined as the ratio of evaporation to effective precipitation. A modified Turc-Pike equation with a horizontal shift is proposed to model interannual variability of seasonal evaporation ratio as a function of seasonal aridity index, which includes rainfall seasonality and soil water change. The performance of the seasonal water balance model is evaluated for 277 watersheds in the United States. The 99% of wet seasons and 90% of dry seasons have Nash-Sutcliffe efficiency coefficients larger than 0.5. The developed seasonal model can be applied for constructing long-term evaporation and storage change data when rainfall, potential evaporation, and runoff observations are available. On the other hand, vegetation affects seasonal water balance by controlling both evaporation and soil moisture dynamics. The correlation between NDVI and evaporation is strong particularly in wet seasons. However, the correlation between NDVI and the seasonal model parameters is only strong in dry seasons.
 Rainfall partition into runoff, evaporation, and soil water storage change and the physical controls of climate, soil, topography, and vegetation on the partition at different temporal and spatial scales are fundamental questions for hydrologists. With the increase of the temporal scale, the complexity of rainfall partition decreases since the temporal variability of hydrologic variables is filtered out in the time-averaged values. Budyko [1958, 1974] postulated that mean annual water balance, represented by the ratio between evaporation and precipitation (E/P), is dominantly controlled by the climate aridity index, which is the ratio between potential evaporation and precipitation (Ep/P). The time scale in the Budyko framework is defined as the long-term average over far more than 1 year [Donohue et al., 2010]. Various functional forms have been developed for quantifying the relation between E/P and Ep/P [Turc, 1954; Pike, 1964; Fu, 1981; Choudhury, 1999; Zhang et al., 2001; Porporato et al., 2004; Yang et al., 2008; Gerrits et al., 2009]. Furthermore, the effects of rainfall seasonality and soil water storage capacity [Milly, 1994a, 1994b; Potter et al., 2005; Hickel and Zhang, 2006; Yokoo et al., 2008; Gerrits et al., 2009; Feng et al., 2012], and vegetation dynamics [Zhang et al., 2001; Donohue et al., 2007] on mean annual water balance have been discussed as a complementary to the climate aridity index. The Budyko framework provides a useful tool to assess the impacts of climate and watershed characteristic changes on annual runoff [Donohue et al., 2011; Roderick and Farquhar, 2011; Wang and Hejazi, 2011; Yang and Yang, 2011].
 The Budyko framework has been applied to interannual variability of rainfall partition in many studies [Koster and Suarez, 1999; Sankarasubramanian and Vogel, 2002; Yang et al., 2007; Potter and Zhang, 2009; Cheng et al., 2011]. Soil water storage changes have been found to be a significant component on the interannual variability of water balance at some study watersheds [Milly and Dunne; 2002; Zhang et al., 2008; Donohue et al., 2010; Istanbulluoglu et al., 2012; Wang, 2012; Tian et al., 2012]. Wang and Alimohammadi  estimated water storage changes as water balance residuals using remote sensing-based evaporation estimations and found that water storage carry-over is significant particularly for watersheds in arid regions. To consider the interannual soil water storage changes in the Budyko framework, Wang  suggested that effective rainfall, which is the difference between rainfall and soil water storage change, is taken as available water supply, and therefore, rainfall in both the climate aridity index and the evaporation ratio is replaced by the computed effective rainfall.
 Both rainfall seasonality and soil water storage change play a significant role on interannual variability of hydrologic responses [Donohue et al., 2012]. Soil water storage capacity, which filters the seasonal rainfall variability, can lower the runoff ratio [Milly, 1993; Sankarasubramanian and Vogel, 2002, 2003; Porporato et al., 2004; Fang et al., 2012]. Zhang et al.  extended the limit concept of Budyko hypothesis to generalized water demand and supply framework, and the framework was applied to the water partition at two stages for developing monthly and daily water balance models. Yokoo et al.  incorporated storage capacity index and drainability index to model water balance at the seasonal scale. Jothityangkoon and Sivapalan  examined the effects of storminess on interannual variability of water balance through the simulation of annual runoff in three semiarid watersheds. Zanardo et al.  studied the within-year rainfall variability controls on annual water balance in a diagnostic and data-driven approach.
 It is necessary to model the interannual variability of rainfall partition at the seasonal scale in order to fully understand the control of rainfall variability and watershed characteristics on interannual water balance. The research question is whether the Budyko hypothesis is applicable for modeling seasonal evaporation when soil water storage changes are significant. The purpose of this paper is to test the Budyko hypothesis at the seasonal scale by defining effective rainfall, which is applied to the definition of monthly and seasonal climate aridity indices and evaporation ratios. Budyko-type functions are modified to model the seasonal evaporation and storage change. The performance of the models is evaluated based on a large number of study watersheds. Given observations of precipitation, runoff, and potential evaporation, the modified Budyko-type equations can be used to estimate seasonal evaporation and storage change which can be further aggregated into annual evaporation and storage carry-over. The modified Budyko-type function at the seasonal scale is introduced in the following section.
2.1. Data Sources
 This study is based on the Model Parameter Estimation Experiment (MOPEX) watersheds with low human interferences [Duan et al., 2006]. Daily precipitation, climatic potential evaporation, and runoff data from 1948 to 2003 are available for the MOPEX watersheds. Daily actual evaporation and monthly potential evaporation from 1983 to 2006 are obtained from the data set provided by University of Montana [Zhang et al., 2010]. Actual evaporation data are derived from remote sensing data and provided at the gridded resolution of 8 km, and the potential evaporation was estimated using Priestley-Taylor method [Priestley and Taylor, 1972] at the same spatial resolution. The daily evaporation and monthly potential evaporation data are spatially averaged to the watershed scale values. This research is focused on the overlapped period of the two data sets from 1983 to 2003. As shown in Figure 1, 277 watersheds, for which there is no missing data during the entire period of 21 years, are selected in this study.
2.2. Quantification of Hydroclimatic Seasonality
 Since the goal of this paper is to model the seasonal evaporation by extending the Budyko hypothesis, wet and dry seasons need to be identified. Seasonality is identified based on monthly data, and a monthly aridity index is introduced to define wet and dry months. Wet and dry seasons can be identified by aggregating wet and dry months, respectively. The definitions of wet and dry months are described in more details in the following section.
2.2.1 Wet and Dry Months
 The monthly aridity index, which follows the concept of climate aridity index, is the ratio of available energy to available water. For long-term water balance, water storage change is usually negligible compared with mean annual precipitation depth. Available energy is represented by potential evaporation, and water availability is represented by precipitation. However, water storage dynamics is significant at the monthly and seasonal scales, and therefore storage change needs to be considered for accounting available water supply. The available water supply in dry months includes not only precipitation but also the depletion of stored water in the watershed, while watershed storage is replenished by infiltrated rainfall in wet months, and the increased storage needs to be subtracted from precipitation. Following Wang , water availability is defined as effective precipitation , and monthly aridity index (Am) is defined as
where EPm and Pm are monthly potential evaporation depth and precipitation depth, respectively; ΔSm is monthly water storage change, including both soil water and groundwater storage changes. Dry and wet months can be defined according to the values of Am: wet months with and dry months with . However, based on this definition, dry months (and wet months) could vary from one year to another due to the interannual variability of monthly aridity index.
 In order to define constant wet and dry months for a given watershed, wet and dry months are identified based on the mean monthly aridity index, which is defined as
where and are averaged monthly potential evaporation depth and precipitation depth over the period of 1983–2003 in this study. Similar with monthly storage change, mean monthly storage changes , are estimated as residuals of the water balance closure , given available data of precipitation, evaporation, and runoff. Months with are identified as dry months, and months with are identified as wet months. As a result, dry and wet months are fixed for a given watershed in this study. Hickel and Zhang  defined storage recharge (i.e., wet) and discharge (i.e., dry) periods based on monthly rainfall and potential evaporation, and the rainfall partitions in the two periods were modeled separately. In this paper, wet and dry months are defined based on the mean monthly aridity index defined by equation ((2)).
2.2.2. Seasonal Aridity Index
 With the wet and dry months identified by equation ((2)), seasonal depths of precipitation, potential evaporation, runoff, and storage change are computed for each year by aggregating monthly values. For example, precipitation depth in the wet season (Pw) and the dry season (Pd) is computed by
where nw and nd are the numbers of wet and dry months in a year and are constants for a given watershed. Similarly, the seasonal values for potential evaporation depth (EPw and EPd), runoff depth (Qw and Qd), and storage changes (ΔSw and ΔSd) are computed based on the monthly values in wet and dry seasons.
 Following the definition of monthly aridity index, seasonal aridity indices for individual years are defined as:
where Am and Ad are the seasonal aridity indices for wet and dry seasons, respectively. Climate seasonality is explicitly modeled in the seasonal aridity index since seasonal rainfall and potential evaporation depths are included in Am and Ad. Seasonal water storage changes in equations ((4a)) and ((4b)) are hydrologic variables, which are controlled by many factors such as soil water storage capacity and infiltration potential. The defined seasonal aridity indices are hydroclimatic variables reflecting both climate seasonality and hydrologic characteristics of watersheds.
 The values of seasonal aridity index for individual years are usually less than 1 for wet seasons and higher than 1 for dry seasons. It should be noted that this may not be valid for all the years, since the definition of dry and wet months is based on the mean monthly aridity index (equation ((2))). If the monthly aridity index for a year deviates significantly from its mean value, it is possible that the seasonal aridity indices are higher than one in wet seasons (or lower than one in dry seasons). It is possible that the mean monthly aridity indices for all 12 months are larger or smaller than 1 for some watersheds where the seasonality is not strong. For these watersheds, there is only one season (wet or dry), and the seasonal aridity index is the exact equivalent of the annual aridity index.
2.2.3. Seasonal Evaporation Ratio
 In the Budyko framework, evaporation ratio is defined as the ratio between actual evaporation and water supply. Following the definition of seasonal aridity index, water supply is represented by the seasonal effective precipitation, and evaporation ratios for wet and dry seasons are modified as and , respectively. In the next section, a Budyko-type function is extended to model the interannual relationship between the seasonal evaporation ratio and the seasonal aridity index defined above.
2.3. Budyko-Type Models at the Seasonal Scale
 The semiempirical equation proposed by Budyko  is a nonparametric model for long-term water balance. To incorporate the effects of other factors on water balance, Budyko-type functions with a single parameter have been developed in the literature [Fu, 1981; Zhang et al., 2001; Yang et al., 2008]. One of the functional forms is the Turc-Pike equation:
where v is the parameter which represents the effects of other factors such as vegetation, soil, and topography on the partition of precipitation. In this paper, the Turc-Pike equation will be extended to model the dependence of the seasonal evaporation ratio on the seasonal aridity index.
 The following two factors are considered in the extension of Budyko-type model to the seasonal scale: (1) the lower bound of the seasonal aridity index for a given watershed and (2) the differentiation between dry and wet seasons. The Budyko equation provides an intercomparison of water balance among watersheds. E/P approaches to zero when climate aridity index approaches to zero in equation ((5)). However, for a given watershed, the lower bound of seasonal aridity index may be a positive value or even higher than 1 in dry seasons. To characterize the possible nonzero lower bound of the seasonal aridity index, a shift along the horizontal axis is introduced to equation (5). On the other hand, two different sets of parameter values in equation (5) are used for wet and dry seasons for the purpose of differentiating the precipitation partitioning behavior in wet and dry conditions.
 As a result, the following modified Turc-Pike equations are proposed to model the seasonal evaporation ratio in wet and dry seasons, respectively:
where vw and vd are the Turc-Pike coefficients in wet and dry seasons, respectively, and ϕw and ϕd are the corresponding lower bounds for the seasonal aridity indices. For the seasonal evaporation model, it is assumed that the functional form of the Budyko curve is applicable to seasonal time scale with the following modifications: (1) seasonal climate aridity index is defined as the ratio of potential evaporation to effective precipitation; (2) seasonal evaporation ratio is defined as the ratio of evaporation to effective precipitation; and (3) the lower bound of seasonal climate aridity index can be more than zero.
 For purposes of demonstration, Figure 2 plots the seasonal evaporation ratio versus seasonal aridity index for four selected watersheds, in which the parameters in equations ((6a)) and ((6b)) are estimated by fitting the observed data points. The Rocky River watershed located in North Carolina (Figure 2a) and the Auglaize River watershed in Ohio (Figure 2b) include both wet (diamond) and dry (circle) seasons. However, the Oostanaula River watershed located in Georgia (Figure 2c) only includes wet seasons, and the Clear Fork Brazos River watershed located in Texas (Figure 2d) only includes dry seasons. As shown in Figure 2, the data points in the wet and dry seasons in Figures 2a and 2b do not follow the same Budyko-type curve. Two separate curves are necessary to model the evaporation ratio for the two seasons, respectively. If there is only one season for a watershed (Figure 2c or 2d), one extended Budyko-type curve is used to model the annual evaporation ratio. Particularly for the Clear Fork Brazos River watershed, which is located in a dry region, the lower bound of seasonal aridity index is more than 2, and a Budyko-type curve with a horizontal shift fits the observations well.
 Two parameters are needed to be estimated in the modified Budyko-type functions for each season. The values of vw and vd represent the physical controls of intraseasonal rainfall (such as storminess) and watershed properties on seasonal evaporation and storage changes. The values of ϕw and ϕd can be interpreted as the lower limits of aridity index for wet and dry seasons. For a given watershed, the value of ϕd should be higher than that of ϕw. Given the same seasonal aridity index in a watershed, the evaporation ratio in dry seasons should be higher than that in wet seasons. The values of ϕw and ϕd also represent the shifts of the 1:1 limit lines due to energy limits. In the seasonal model of Hickel and Zhang , when mean monthly rainfall exceeds potential evaporation during wet seasons, evaporation is assumed to occur at the potential rate for enabling a minimum-parameter formulation. The effect of this assumption appears to be minimal since they focus on mean annual water balance. However, this study focuses on the seasonal variability of evaporation and storage change, so the evaporation in wet seasons is modeled by equation ((6a)). When a seasonal aridity index is smaller than 1 in the wet season, the upper bound of evaporation is equal to , which is usually smaller than EPw. On the other hand, in dry seasons with , the upper limit of Ed is , which is smaller than the water supply . As a result, there is a smaller upper bound on seasonal evaporation in “energy-limited” conditions.
2.4. Modeling Annual Storage Changes
 Once the four parameters (vw, vd, ϕw, and ϕd) for the seasonal evaporation model are obtained, the seasonal Budyko-type model developed in this paper can be used to estimate annual storage changes and evaporation if precipitation, potential evaporation, and runoff observations are available. Substituting into equations ((6a)) and ((6b)), the following equations are obtained and can be used to estimate storage changes in wet and dry seasons:
 The values of ΔSw and ΔSd can be solved numerically using equations ((7a)) and ((7b)), and annual storage changes (ΔS) can be computed as a summation of seasonal storage changes:
 The annual evaporation can be computed as a residual of water balance once storage changes are estimated.
2.5. Model Performance Evaluation
 The model performance is evaluated using two indicators: root-mean-square error (RMSE) and Nash-Sutcliffe efficiency (here referred to as coefficient of efficiency (CE)). RMSE is calculated as
where Xo,i and Xm,i are the observed and modeled values in the ith year, respectively; n is the number of years. CE shows the extent to which observed and modeled values follow the line with 1:1 slope [Moriasi et al., 2007]. CE is calculated as
 CE ranges from −∞ to 1. Values close to 1 indicate higher model efficiency in predicting actual values [Legates and McCabe, 1999]. A positive CE value is usually acceptable for a model [Moriasi et al., 2007].
 RMSE and CE are applied to evaluate the fitness of the extended Budyko-type model and the performance of the model in estimating annual storage changes from equations ((7a)), ((7b)), and ((8)). The fitness of the seasonal Budyko-type model is computed for all the watersheds in each season and is compared among watersheds.
3. Results and Discussions
 The developed model in this paper is applied to the 277 case study watersheds shown in Figure 1. Based on the definition of wet and dry months, 203 watersheds have both wet and dry seasons and 191 watersheds have consecutively dry months in summer seasons. The duration of dry seasons ranges from 1 to 11 months in these watersheds. Fifty-one watersheds only have wet seasons, and most of them are located in the northeastern corner of the United States and the Appalachian Mountain area. Twenty-three watersheds only have dry seasons and most of them are located in the High Plains.
 The seasonal model based on the Turc-Pike equation is fitted to the observations for each watershed. The estimated parameter values for the study watersheds are then discussed. Based on the estimated parameters and equations ((7a)), ((7b)), and ((8)), annual storage changes are computed and the performance of the model is evaluated. The vegetation controls on evaporation and the four model parameters in equations ((6a)) and ((6b)) are discussed. At the end, the uncertainty of evaporation data on the model performance is assessed.
3.1. Storage Change Impact on Interannual Water Balance
 One of the objectives of this paper is to model interannual storage changes by aggregating seasonal variables. The impact of storage change from year to year on the representation of Budyko hypothesis is assessed for the study watersheds. Figure 3 presents the water balance in the annual scale of all the study watersheds in the Budyko's framework with three different computations of aridity index or evaporation ratio. In Figure 3a, evaporation is estimated as the difference between precipitation and runoff. This representation is usually used when evaporation data are not available. Figure 3b represents E/P versus Ep/P. Such approach to describe interannual water balance was presented by Cheng et al. . As shown in Figure 3b, if P is considered as water supply in the annual scale, E/P is higher than 1 in many cases. The uncertainty of E may contribute to this but is not enough to explain the high evaporation in extreme dry years. This result highlights the fact that available water supply is not limited to precipitation only, but storage changes also play a significant role in maintaining evaporation, especially for years with aridity indices higher than 1. Figure 3c shows the plot of E/(P − ΔS) versus Ep/(P − ΔS) when P − ΔS is used to represent available water instead of P. From this comparison, it can be interpreted that the Budyko hypothesis is applicable at the interannual scale, if the supply of energy and water are described accurately.
3.2. Application of the Seasonal Model to Case Study Watersheds
3.2.1. Performance of the Modified Seasonal Turc-Pike Model
 The developed seasonal model based on the Budyko-type function in equations ((6a)) and ((6b)) is applied to the case study watersheds shown in Figure 1. The values of the four seasonal parameters (vw, vd, ϕw, and ϕd) are estimated based on the available data for monthly precipitation, potential evaporation, evaporation, and runoff during 1983–2003. For example, Figure 2 shows the modified Turc-Pike curves in wet and dry seasons that fit to the data points for 4 watersheds from the 277 case study watersheds. As shown in Figure 2a for the Rocky River watershed, parameters in wet seasons are estimated as ϕw = 0.13 and vw = 2.40, and parameters in dry seasons are estimated as ϕd = 0.14 and vd = 7.39. As shown in Figure 2b for the Auglaize River watershed, wet season parameters are estimated as ϕw = 0.16 and vw = 1.34, and dry season parameters are ϕd = 0.26 and vd = 6.10. To evaluate the performance of the model, CE values are calculated for the Rocky River watershed and the Auglaize River watershed. The CE values for the estimated seasonal evaporation ratio in wet seasons are 0.98 and 0.97 for the two watersheds, respectively, and the CE values in dry seasons are 0.96 and 0.90. Figure 2c shows a fitted curve for the Oostanaula River watershed in which all the 12 months are classified as wet seasons, and the value of CE is 0.99. The estimated values are 0.11 and 3.19 for ϕw and vw, respectively. The Clear Fork Brazos River watershed in Figure 2d only includes the dry seasons and the values of ϕd and vd for the fitted curve are 2.44 and 4.89, with a CE value of 0.67.
 To evaluate the overall performance of the model, the frequency distribution of CE for all 277 case study watersheds was calculated and is presented in Figure 4. In wet seasons (Figure 4a), CE values in 99% of watersheds are higher than 0.5, and CE values in 81% of watersheds are higher than 0.9. In dry seasons (Figure 4b), CE values in 90% of watersheds are higher than 0.5, and CE values in 40% of watersheds are higher than 0.9. The model performance in wet seasons is generally better than that in dry seasons. The number of watersheds at the peak frequency is 139 with CE value around 0.925–0.975 in wet seasons (Figure 4a); while the number of watersheds at the peak frequency is 59 with CE value of 0.875–0.925 in dry seasons (Figure 4b). In general, the seasonal model in equations ((6a)) and ((6b)) works very well for the interannual water balance at the seasonal scale.
3.2.2. Estimated Model Parameters
 In the seasonal model, the evaporation ratio is a function of the seasonal aridity index and the parameters vw and ϕw in wet seasons or vd and ϕd in dry seasons. The values of the parameters reflect the dependence of seasonal evaporation and storage changes on other factors such as intraseasonal rainfall, vegetation, soil properties, and topography in the watershed. Figure 5 shows the histograms of the four parameters (Figure 5a for the shift parameter ϕw in wet seasons, Figure 5b for the Turc-Pike parameter vw, Figure 5c for ϕd, and Figure 5d for vd). The values of ϕw have the highest frequency around 0.1, while values of ϕd have the highest frequency around 0.25. This is due to the higher value of minimum aridity index in dry seasons compared with wet seasons. The values of vw have the highest frequency around 1.5, though, in some cases, values higher than 10 were observed; values of vd have the highest frequency around 5. The value of vd is usually larger than that of vw for a given watershed. The parameter values of v in dry seasons are more dispersed compared with those in wet seasons.
3.2.3. Vegetation Control on Seasonal Evaporation Ratios
 Climate seasonality and vegetation adaption controls on annual water balance have been one of the focused research areas in recent years [Feng et al., 2012; Gentine et al., 2012; Xu et al., 2012]. Vegetation control on seasonal evaporation and storage change is explored in wet and dry seasons separately in this paper. Normalized Difference Vegetation Index (NDVI) is used as a proxy for vegetation. Bimonthly NDVI data based on the Advanced Very High Resolution Radiometer (AVHRR) imagery from the Global Inventory Modeling and Mapping Studies (GIMMS) can be downloaded at http://glcf.umiacs.umd.edu/data/gimms/ [Tucker et al., 2005]. Averaged values of NDVI at the monthly and seasonal scales are computed for each of the study watersheds.
 Vegetation affects the seasonal water balance through both evaporation and soil moisture dynamics. Strong correlations exist between monthly average NDVI and evaporation. The percentage of watersheds where the correlation coefficients (r) between monthly NDVI and evaporation are higher than 0.5 is 96% in wet seasons and 73% in dry seasons. To quantify the potential interaction between vegetation and evaporation in wet and dry seasons, a bivariate Granger causality test [Granger, 1969; Engle and Granger, 1987; Detto et al., 2012] is conducted between monthly NDVI and evaporation. A 10% significance level is used in the Granger test. In dry seasons, evaporation is the cause and NDVI is the effect in 71% of the watersheds, and NDVI is the cause and evaporation is the effect in 59% of the watersheds. In wet seasons, evaporation is the cause and NDVI is the effect in 92% of the watersheds, and NDVI is the cause and evaporation is the effect in 81% of the watersheds. These results on the Granger causality test show the interaction and feedback between vegetation and evaporation.
 Vegetation controls seasonal water balance not only by evaporation but also by soil moisture dynamics. In the developed seasonal model of equation (6), seasonal storage changes have been included into the seasonal aridity index. The controls of other factors such as vegetation, rainfall intensity, and infiltration capacity are reflected by the parameters, and the corresponding controls may be different with wet and dry seasons. To evaluate the vegetation control on seasonal water balance, Figure 6 plots the dependence of ϕd, vd, ϕw, and vw as a function of long-term average seasonal NDVI values for all 277 watersheds. Strong correlation between NDVI and dry season parameters is identified. As shown in Figure 6a, when NDVI is smaller than 0.5, ϕd is not sensitive to NDVI (r = −0.273). The absolute value of correlation coefficient between NDVI and ϕd increases when NDVI is larger than 0.5 (r = −0.679). As discussed earlier, ϕd corresponds to the lower bound of the dry season aridity index. According to Figure 6a, watersheds with higher NDVI have lower bounds of aridity index in dry seasons. This is due to the fact that higher vegetation coverage has a greater potential to deplete soil water storage during drought periods, which in turn induces smaller values of the dry season aridity index, . As shown in Figure 6b, vd increases with NDVI and the correlation coefficient between NDVI and vd is 0.557. Higher values of vd correspond to higher evaporation ratios, . However, the relationships between NDVI and the wet season parameters are nonmonotonic as shown in Figures 6c and 6d. The correlation coefficient is −0.24 in Figure 6c and 0.01 in Figure 6d, respectively. It seems that a maximum value of ϕw occurs around NDVI = 0.4.
3.2.4. Estimation of Annual Storage Changes
 As mentioned before, once the values of parameters for each watershed are estimated, the seasonal model developed in this paper can be used to estimate annual evaporation and storage changes when precipitation, potential evaporation and runoff data are available. Storage changes are estimated by equations ((7a)) and ((7b)) for wet and dry seasons, which are then aggregated to annual storage changes by equation (8). The model's performance on modeling storage changes is evaluated by dividing the historical data into calibration (1983–1992) and validation (1993–2002) periods. The four parameters in equations ((6a)) and ((6b)) are estimated based on observations during the calibration period. The annual storage changes during the validation period are computed and compared with the “observed” annual storage changes estimated by water balance closure. The comparison is presented in Figure 7: Figure 7a for watersheds with both wet and dry seasons, Figure 7b for watersheds with dry seasons only, and Figure 7c for watersheds with wet seasons only. In Figure 7a, the average RMSE is 27 mm for dry seasons and 21 mm for wet seasons. The average value of RMSE is 54 mm for Figure 7b and 18 mm for Figure 7c. The overall average RMSE of annual storage changes for these 277 watersheds is 24 mm. The performance in wet seasons is better than in dry seasons, especially when comparing wet season only watersheds to dry season only watersheds.
3.2.5. Impacts of Evaporation Data Uncertainty
 The uncertainties in observations, particularly evaporation estimation from remote sensing data, may contribute to the unrealistic storage change and further decrease the performance of the extended seasonal Budyko model. The observed storage changes are up to 800 mm in a few watersheds as shown in Figure 7 and this may be unrealistic. To evaluate the impacts of evaporation data uncertainty on the results, 158 watersheds from the total 277 watersheds discussed by Wang and Alimohammadi , where the difference of long-term average annual evaporation between remote sensing-based and water balance-based estimation is within ±10%, are selected for further investigation. The magnitude of observed annual storage changes in the 158 watersheds decreases significantly and the storage change values range from −400 mm to 400 mm. The average value of CE over the 277 watersheds is 0.958 for wet seasons and 0.878 for dry seasons (Figure 4). The average value of CE over the 158 watersheds increases to 0.968 for wet seasons and 0.882 for dry seasons. It indicates that the impact of the evaporation data uncertainty is not very significant on the seasonal model performance.
3.3. Physically Based Processes Versus Coevolution
 The Budyko hypothesis on mean annual water balance results from the coevolution of watershed vegetation, soil, and geomorphology with climate [Gentine et al., 2012; Troch et al., 2013; Wang and Wu, 2013]. As demonstrated in Figure 8, the strength of coevolution (Darwinian view) will become weaker with reducing time scales, and physical processes-based models (Newtonian view) for evaporation will take over at the small time scale (e.g., daily). Harman and Troch  review the success of Darwinian method in hydrologic science and call for synthesis of the Darwinian and Newtonian approaches as a remaining goal. Great progresses are expected if the Newtonian approach can be reconciled with the Darwinian view [Sivapalan, 2005; Troch et al., 2013]. One purpose of this work is to assess the strength of coevolution view, presented by Budyko framework, on modeling evaporation at the shorter time scale. Figure 9 shows the monthly evaporation ratio versus monthly aridity index for the four watersheds shown in Figure 2. From seasonal to monthly scale, CE values decrease from 0.98 to 0.90 (wet) and 0.97 to 0.84 (dry) for Rocky River watershed, from 0.98 to 0.64 (wet) and 0.95 to 0.46 (dry) for Auglaize River watershed. CE values for Oostanaula River watershed decrease from 0.99 to 0.92 at all the wet months; particularly CE values for Clear Fork Brazos River decrease from 0.68 to −2.09 for all the dry months. The performance of the extended Turc-Pike equation declines significantly from seasonal to monthly scales. Therefore, the strength of Darwinian approach for modeling evaporation may be not compelling at the monthly scale.
4. Summary and Future Work
 Rainfall partitions from long-term to interannual and to seasonal scales are important research issues in the hydrologic sciences community. Climate is the first-order control on mean water balance at the annual scale, and vegetation and the interaction between climate seasonality and soil water storage change have also been found to play important roles. At the seasonal scale, the effects of storminess, infiltration capacity, and topography emerge, and soil water storage becomes more significant. Following the Budyko hypothesis, a new seasonal aridity index is defined as the ratio of potential evaporation depth to effective precipitation depth, which accounts for the storage changes in water supply. Similarly, a new evaporation ratio is defined as the ratio of evaporation depth to effective precipitation depth. A modified Budyko-type model is proposed to model the interannual variability of the seasonal evaporation ratio as a function of the seasonal aridity index. The seasonal values are aggregated to quantify the interannual variability of evaporation and storage changes. Rainfall seasonality and seasonal soil water storage dynamics are incorporated into the developed seasonal model directly, which facilitates the understanding of the dominant controlling factors on water balance from mean annual to seasonal scales.
 The performance of the seasonal water balance model is evaluated using data from 277 watersheds, where daily rainfall, runoff, and both remote-sensing based evaporation and monthly potential evaporation data are available. Based on the long-term mean monthly aridity index, wet and dry seasons are identified for each watershed. A Turc-Pike equation with a horizontal shift is fitted to the observed seasonal evaporation ratio versus the seasonal aridity index. The seasonal model works well, and the performance in wet seasons is better than that in dry seasons. Once the seasonal parameters are determined, the seasonal model can be applied to estimate long-term seasonal evaporation and storage changes if rainfall, potential evaporation, and runoff observations are available. If runoff observations are not available, an additional equation describing the dependence of runoff on soil water storage is needed to capture the seasonal hydrologic dynamics. Therefore, future work will develop seasonal runoff equations to complete the water balance closure at the seasonal scale.
 Physical controls of climate, rainfall characteristics, soil, vegetation, and topography on mean annual and interannual water balance have been studied in the past decades. The individual factor may have different effects in dry and wet seasons. In this paper, strong correlation between vegetation and dry season parameters has been found but correlation between vegetation and wet season parameters is weak. Future work will focus on the controls of other factors on seasonal evaporation and storage changes.
 This research was funded in part under Award NA10OAR4170079 from Florida Sea Grant and Award NA10NOS4780146 from the National Oceanic and Atmospheric Administration (NOAA) Center for Sponsored Coastal Ocean Research (CSCOR). The authors are grateful to the insightful comments, which led to improvement of this manuscript, by three anonymous reviewers and the Associate Editor.