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

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## Abstract

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_{P})) 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 ET

_{p}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 streamflow modified the impact of climate controls on mean annual water balance in the baseflow-dominated watersheds.

## 1. Introduction

_{p})) 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.

## 2. Materials and Methods

#### 2.1. Data Used

_{max}) and minimum daily temperature (averaged over all days in the month) (T

_{min}), and solar radiation (R

_{a}). 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 BFI

_{max}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.

#### 2.2. Potential Evapotranspiration Calculation

_{P}in previous studies [31,32,33]. The equation can be expressed as [30]:

_{p}is the potential evapotranspiration, mm/d; T

_{max}and T

_{min}are the maximum monthly temperature and minimum monthly temperature, respectively, °C; R

_{a}is the extraterrestrial solar radiation, mm/d.

_{p}in mm/d using Equation (1). Annual ET

_{p}in the study watersheds was derived from summing monthly ET

_{p}of each year.

Gaging Station ID | Station Name and Location | Latitude | Delineated Area (km^{2}) | P (mm/yr) | ET_{P} (mm/yr) | Q (mm/yr) | ET_{a} (mm/yr) | $\frac{{\text{ET}}_{\text{p}}}{\text{P}}$ | Q/P | $\frac{{\text{ET}}_{\text{a}}}{\text{P}}$ | BFI |
---|---|---|---|---|---|---|---|---|---|---|---|

04040500 | Sturgeon River near Sidnaw | 46.584 | 429.8 | 878 | 832 | 416 | 462 | 0.95 | 0.47 | 0.53 | 0.66 |

04043050 | Trap Rock River near Lake Linden | 47.229 | 77.1 | 807 | 757 | 529 | 278 | 0.94 | 0.66 | 0.34 | 0.66 |

04045500 | Tahquamenon River near Paradise | 46.575 | 1960.6 | 828 | 832 | 395 | 432 | 1.00 | 0.48 | 0.52 | 0.73 |

04057510 | Sturgeon River near Nahma Junction | 45.943 | 475.4 | 826 | 866 | 348 | 478 | 1.05 | 0.42 | 0.58 | 0.74 |

04059500 | Ford River near Hyde | 45.755 | 1156.8 | 776 | 843 | 278 | 499 | 1.09 | 0.36 | 0.64 | 0.68 |

04096405 | Sturgeon River at Wolverine | 45.274 | 454.7 | 850 | 906 | 402 | 449 | 1.07 | 0.47 | 0.53 | 0.80 |

04105000 | Manistee River near Sherman | 44.436 | 2241.9 | 835 | 912 | 428 | 407 | 1.09 | 0.51 | 0.49 | 0.80 |

04105700 | Pere Marquette River at Scottville | 43.945 | 1787.7 | 883 | 963 | 396 | 487 | 1.09 | 0.45 | 0.55 | 0.79 |

04108600 | Macatawa River at State Road near Zeeland | 42.779 | 172.9 | 930 | 951 | 406 | 525 | 1.02 | 0.44 | 0.56 | 0.45 |

04108800 | Rabbit River near Hopkins | 42.642 | 174.9 | 938 | 934 | 307 | 631 | 1.00 | 0.33 | 0.67 | 0.70 |

04117500 | Thornapple River near Hastings | 42.616 | 1063.4 | 886 | 972 | 328 | 558 | 1.10 | 0.37 | 0.63 | 0.71 |

04122500 | Augusta Creek Near Augusta | 42.353 | 95.2 | 949 | 1003 | 394 | 555 | 1.06 | 0.42 | 0.58 | 0.78 |

04124000 | Battle Creek at Battle Creek | 42.331 | 710 | 888 | 970 | 335 | 552 | 1.09 | 0.38 | 0.62 | 0.72 |

04127997 | St. Joseph River at Burlington | 42.103 | 530.7 | 932 | 957 | 319 | 613 | 1.03 | 0.34 | 0.66 | 0.76 |

04161580 | Stony Creek near Romeo | 42.801 | 61.7 | 826 | 925 | 237 | 589 | 1.12 | 0.29 | 0.71 | 0.69 |

04164000 | Clinton River near Fraser | 42.578 | 1188.3 | 823 | 943 | 340 | 483 | 1.15 | 0.41 | 0.59 | 0.70 |

04166100 | River Rouge at Southfield | 42.448 | 225.3 | 815 | 954 | 307 | 509 | 1.17 | 0.38 | 0.62 | 0.61 |

#### 2.3. Water Balance Modeling Based on Budyko Hypothesis

_{a}is primarily controlled by available water and energy. Budyko Hypothesis is a model that represents ET

_{a}/P ratio as a function of climatic aridity index. It can be expressed as:

_{a}/P is the ET

_{a}ratio; ET

_{p}/P is the climate aridity index; F is an empirical function.

_{p}are the dominant factors controlling mean annual ET

_{a}. Budyko’s Equation can be represented as [4]:

_{a}was derived from water balance equation (i.e., ET

_{a}= 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:

_{obs,i}is the observed mean annual streamflow (mm/year); Q

_{pred,i}is the predicted mean annual streamflow (mm/year); n is the total number of the study watersheds (n = 17).

#### 2.4. Climate Sensitivity of Streamflow Based on the Budyko Hypothesis

_{p}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 ET

_{p}) 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 ET

_{p}on variation in mean annual Q can be expressed as [8,36]:

_{P}) represent Q sensitivity to a 1-unit variation in mean annual P and ET

_{p}. Larger ∂Q/∂P and ∂Q/∂ET

_{P}values represent greater influence of variation in mean annual P and ET

_{p}on Q.

_{p}were computed from Fu’s Equation (Equation (5)). The mathematical expressions for computing ∂Q/∂P and ∂Q/∂ ET

_{P}can be expressed as [8]:

_{P}was computed as [8]:

_{P}changes, respectively.

_{P}. For example, sensitivity coefficients of Q responses to P and ET

_{P}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 ET

_{P}decreased Q by 5%.

## 3. Results and Discussion

#### 3.1. Mean Annual Water Balance in the Study Watersheds

_{p}/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 ET

_{p}/P ranges. Xu et al. [38] also reported similar ET

_{p}/P ratios, which vary from 0.87 to 1.33 in 55 watersheds across the Midwest United States. Mean annual ET

_{a}/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 ET

_{a}/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 ET

_{a}/P ratios were low. As shown in Figure 1, the relationship between mean annual ET

_{a}/P and ET

_{p}/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).

**Figure 1.**Comparison of observed and calculated (using Fu’s Equation) ET

_{a}/P in the study watersheds.

#### 3.2. Impact of Climatic Controls on Mean Annual Water Balance

_{p}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 ET

_{p}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 ET

_{p}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 ET

_{p}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.

**Table 2.**Sensitivity of mean annual Q to variations in mean annual P and ET

_{p}in the study watersheds in Michigan.

Gaging Station ID | ∂Q/∂P | ∂Q/∂ET_{p} | (P/Q) × (∂Q/∂P) | (ET_{p}/Q) × (∂Q/∂ET_{p}) |
---|---|---|---|---|

04040500 | 0.74 | −0.29 | 1.57 | −0.58 |

04043050 | 0.83 | −0.19 | 1.26 | −0.27 |

04045500 | 0.73 | −0.26 | 1.53 | −0.54 |

04057510 | 0.69 | −0.26 | 1.64 | −0.66 |

04059500 | 0.65 | −0.27 | 1.80 | −0.83 |

04096405 | 0.66 | −0.31 | 1.92 | −0.93 |

04105000 | 0.66 | −0.26 | 1.75 | −0.76 |

04105700 | 0.69 | −0.26 | 1.66 | −0.67 |

04108600 | 0.65 | −0.33 | 2.00 | −1.02 |

04108800 | 0.71 | −0.27 | 1.62 | −0.63 |

04117500 | 0.66 | −0.26 | 1.77 | −0.78 |

04122500 | 0.70 | −0.24 | 1.56 | −0.58 |

04124000 | 0.74 | −0.21 | 1.44 | −0.45 |

04127997 | 0.72 | −0.24 | 1.52 | −0.54 |

04161580 | 0.59 | −0.28 | 2.06 | −1.09 |

04164000 | 0.67 | −0.23 | 1.62 | −0.64 |

04166100 | 0.64 | −0.23 | 1.70 | −0.72 |

## 4. Discussions

#### 4.1. Compared with Other Similar Studies

_{a}/P ratios varied across the study watersheds. Previous studies suggested that differences in ET

_{a}/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 ET

_{p}/P), with ET

_{a}/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 ET

_{a}/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.

_{a}/P ratios (e.g., Thornapple River and Battle Creek) (Table 1). Inversely, if BFI values were different, ET

_{a}/P ratios would be different. Although average annual water balance was principally controlled by available water and energy (i.e., P and ET

_{p}), 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.

#### 4.2. Limitations

_{p}is an important variable for estimating ET

_{a}and climate aridity index in hydrological modeling. Limited to the data availability, the Hargreaves method [30] was selected to calculate ET

_{p}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 ET

_{p}estimates can be improved by adding more relevant input variables [46,47]. Thus, the impact of ET

_{p}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 ET

_{a}/P and ET

_{p}/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.

## 5. Conclusions

_{p}) 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 ET

_{a}/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 ET

_{p}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 ET

_{p}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.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Zhang, Y.; Engel, B.; Ahiablame, L.; Liu, J.
Impacts of Climate Change on Mean Annual Water Balance for Watersheds in Michigan, USA. *Water* **2015**, *7*, 3565-3578.
https://doi.org/10.3390/w7073565

**AMA Style**

Zhang Y, Engel B, Ahiablame L, Liu J.
Impacts of Climate Change on Mean Annual Water Balance for Watersheds in Michigan, USA. *Water*. 2015; 7(7):3565-3578.
https://doi.org/10.3390/w7073565

**Chicago/Turabian Style**

Zhang, Yinqin, Bernard Engel, Laurent Ahiablame, and Junmin Liu.
2015. "Impacts of Climate Change on Mean Annual Water Balance for Watersheds in Michigan, USA" *Water* 7, no. 7: 3565-3578.
https://doi.org/10.3390/w7073565