# Historical Trends in Mean and Extreme Runoff and Streamflow Based on Observations and Climate Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. Hydrological Model (WBM) Performance Evaluation

#### 3.2. Runoff and Streamflow Trends 1971–2001 (GCMs versus Observations)

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Mann-Kendall Trend Test

_{i}and x

_{j}are the data values in time series i and j (j > i), respectively, and sgn(x

_{j}− x

_{i}) is the sign function:

_{i}is the number of ties of extent i. A tied group is a set of sample data having the same value. In cases where the sample size n > 10, the standard normal test statistic Z

_{S}is computed as:

_{S}indicates the trend in the data series, where positive values of Z

_{S}means increasing trend, while negative Z

_{S}values show decreasing trends. For the tests at a specific α significance level, if $\left|{Z}_{\mathrm{S}}\right|>{Z}_{1-\mathsf{\alpha}/2}$, the null hypothesis is rejected and the time series has a statistically significant trend. ${Z}_{1-\mathsf{\alpha}/2}$ is obtained from the standard normal distribution table, where at the 5% significance level (α = 0.05), trend is statistically significant if $\left|{Z}_{\mathrm{S}}\right|>1.96$ and at the 1% significance level (α = 0.01), trend is statistically significant if $\left|{Z}_{\mathrm{S}}\right|>2.576$.

## Appendix B. Sen’s Slope Estimator

_{j}and x

_{k}are the data values at times j and k (j > k), respectively. N is defined as $\frac{n\left(n-1\right)}{2}$, where n is the number of time periods.

_{i}are ranked from smallest to largest, the parameter Qmed is computed as the median of the Q

_{i}vector. The Qmed sign reflects the direction of trend, while its value indicates the magnitude of the trend. To determine whether the median slope is statistically different than zero, the confidence interval of Qmed at a specific probability should be computed as follows [58,59]:

_{1}th largest and the (M

_{2}+ 1)th largest of the N ordered slope estimates [58]. The slope Qmed is significantly different than zero if the two limits Qmin and Qmax have the same sign.

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**Figure 1.**WBM model versus observations streamflow data 1971–2001 maps for Coterminous US—Log of annual-averaged daily streamflow map for observations (

**a**) and WBM model (

**d**), Slope of change in streamflow map for observations (

**b**) and WBM model (

**e**) and relative change in annual-averaged daily streamflow map for observations (

**c**) and WBM model (

**f**). Lower axis of the maps represents the latitude and left side axis represents the longitude of the girds.

**Figure 2.**WBM model versus observations streamflow data 1971–2001 boxplots for Coterminous US—Log of annual-averaged daily streamflow map for observations (

**a**), Slope of change in streamflow map for observations (

**b**) and relative change in annual-averaged daily streamflow map for observations (

**c**). Values larger than 99

^{th}and smaller than 1

^{st}percentile of the data (considered as the outliers) have been excluded from the plots. The boxplots show good agreement between the observations and WBM model outputs for the relative change in streamflow.

**Figure 3.**Global maps of runoff simulations of WBM model from WFD observation-based input versus ISI-MIP model simulations input, 1971–2001—Mean runoff map for WFD (

**a**) and ISI-MIP (

**d**), Slope of change in mean runoff map for WFD (

**b**) and ISI-MIP (

**e**), and relative change in mean runoff map for WFD (

**c**) and ISI-MIP (

**f**). Lower side axes of the maps represent the latitudeand left side axes represent the longitude of the girds.

**Figure 4.**Global maps of WBM streamflow generated from WFD observation-based input versus ISI-MIP inputs (average of the 5 GCMs), 1971–2001—mean streamflow map for WFD (

**a**) and ISI-MIP (

**d**), Slope of change in mean streamflow map for WFD (

**b**) and ISI-MIP (

**e**), and relative change in mean streamflow map for WFD (

**c**) and ISI-MIP (

**f**). The values are per grid cell.

**Figure 5.**Boxplots of runoff and streamflow trends simulated under climate forcings from either ISI-MIP models (minimum, 25th percentile, median, 75th percentile and maximum of the 5 model runs) or WFD observation-based data (shown as blue circles) for 1971–2001 on global and continental scale—(

**a**) annual average of runoff (mm·day

^{−1}), (

**b**) linear regression slope of change in annual-averaged runoff (mm·day

^{−1}·year

^{−1}), (

**c**) relative change in annual-averaged runoff (%·year

^{−1}), (

**d**) average of discharge (m

^{3}·s

^{−1}), (

**e**) linear regression slope of change in discharge (m

^{3}·s

^{−1}·year

^{−1}), (

**f**) relative change in discharge (%·year

^{−1}).

**Figure 6.**ISI-MIP global map of relative change in extreme runoff (

**a**) and extreme streamflow (

**d**). Boxplots of trends simulated under climate forcing from ISI-MIP models (minimum, 25th percentile, median, 75th percentile and maximum of the 5 model runs) for relative change in annual-maximum 1-day runoff (%·year

^{−1}) (

**b**), annual-maximum 1-day streamflow (%·year

^{−1}) (

**e**) and annual-maximum 7-day streamflow (%·year

^{−1}) (

**c**).

**Table 1.**Statistics of variation of streamflow trend results for observations, as well as WBM simulations in the Coterminous United States from 1971 to 2001. The observational discharge is station measurements; whereas the WBM simulated discharge average is per grid cell. WBM results are for all grid cells of the Coterminous US, as shown in Figure 1d. Ave. = Average, Min. = Minimum, Max. = Maximum, Med. = Median, St. Dev. = Standard Deviation.

Discharge Ave (Q) (m^{3}·s^{−1}) | Slope of Change (b) (m^{3}·s^{−1}·year^{−1}) | Relative Change (b/Q) (%·year^{−1}) | ||||
---|---|---|---|---|---|---|

Obs. | WBM | Obs. | WBM | Obs. | WBM | |

Grids Ave. | 106.4 | 217.7 | −0.16 | −0.64 | −0.01 | −0.35 |

Grids Min. | 0.0 | 0.0 | −7.81 | −26.19 | −6.48 | −12.01 |

Grids Max. | 1776.2 | 11332.9 | 7.66 | 23.74 | 6.87 | 11.81 |

Grids Med. | 42.6 | 35.1 | −0.04 | −0.07 | −0.28 | −0.33 |

Grids St. Dev. | 199.5 | 810.3 | 1.40 | 3.13 | 1.69 | 2.40 |

**Table 2.**Global and continental-averaged trend results of the WBM runoff simulations based on the WFD observation-based inputs, from 1971 to 2001. The Qmed and Z-score indices are obtained from the Sen’s slope estimator and Mann-Kendall methods, respectively, for the runoff.

Runoff Ave (R) (mm·day^{−1}) | Slope of Change (b) (mm·day^{−1}·year^{−1}) | Relative Change (b/R) (%·year^{−1}) | Qmed (mm·day^{−1}·year^{−1}) | Z Score (-) | |
---|---|---|---|---|---|

Global | 0.23 | −0.00038 | −0.042 | −0.00035 | −0.05 |

North America | 0.88 | −0.00211 | −0.307 | −0.00216 | −0.37 |

South America | 1.95 | −0.00572 | −0.355 | −0.00560 | −0.42 |

Europe | 0.74 | 0.00104 | 0.211 | 0.00125 | 0.23 |

Oceania | 0.42 | −0.00150 | −0.597 | −0.00071 | −0.27 |

Africa | 0.89 | −0.00077 | 0.009 | −0.00069 | −0.11 |

Asia | 0.96 | −0.00086 | −0.186 | −0.00085 | −0.25 |

India | 1.26 | −0.00351 | −0.758 | −0.00290 | −0.31 |

**Table 3.**Global-averaged results of mean and annual-maximum 1-day runoff trend analysis—WBM runoff simulations based on the WFD observation-based inputs as well as from the ISI-MIP (WBM model driven by GCM climate forcing), for 1971–2001. The 5 ISI-MIP models give 5 global averages, of which the minimum, maximum, median, mean, and standard deviation are presented. Ave. = Average, Min. = Minimum, Max. = Maximum, Med. = Median, St. Dev. = Standard Deviation.

Runoff Ave (R) (mm·day^{−1}) | Slope of Change (b) (mm·day^{−1}·year^{−1}) | Relative Change (b/R) (%·year^{−1}) | Qmed (mm·day^{−1}·year^{−1}) | Z Score (-) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean Runoff | Max. 1d Runoff | Mean Runoff | Max. 1d Runoff | Mean Runoff | Max. 1d Runoff | Mean Runoff | Max. 1d Runoff | Mean Runoff | Max. 1d Runoff | |

WFD | 0.23 | - | −0.00038 | - | −0.042 | - | −0.00035 | - | −0.05 | - |

ISI-MIP Ave. | 0.22 | 2.95 | 0.00005 | 0.00399 | 0.031 | 0.035 | 0.00000 | 0.00211 | 0.011 | 0.019 |

ISI-MIP Min. | 0.21 | 2.69 | −0.00013 | 0.00006 | −0.012 | −0.010 | −0.00019 | −0.00070 | −0.007 | 0.003 |

ISI-MIP Max. | 0.22 | 3.24 | 0.00025 | 0.00727 | 0.061 | 0.062 | 0.00015 | 0.00419 | 0.038 | 0.039 |

ISI-MIP Med. | 0.21 | 2.95 | −0.00001 | 0.00465 | 0.036 | 0.036 | −0.00005 | 0.00205 | 0.004 | 0.015 |

ISI-MIP St. Dev. | 0.00 | 0.21 | 0.00016 | 0.00283 | 0.026 | 0.029 | 0.00015 | 0.00181 | 0.019 | 0.016 |

**Table 4.**Global and continental-averaged trend results of the WBM streamflow simulations based on the WFD observation-based inputs, from 1971 to 2001. The average discharge magnitude (Q) values are per grid cell. The Qmed and Z-score indices are obtained from the Sen’s slope estimator and Mann-Kendall methods, respectively, for the streamflow.

Discharge Ave (Q) (m^{3}·s^{−1}) | Slope of Change (b) (m^{3}.s^{−1}·year^{−1}) | Relative Change (b/Q) (%·year^{−1}) | Qmed (m^{3}·s^{−1}·year^{−1}) | Z Score (-) | ||
---|---|---|---|---|---|---|

Global | 116.50 | −0.28 | −0.041 | −0.29 | −0.06 | |

North America | 262.53 | −0.86 | −0.281 | −0.89 | −0.41 | |

South America | 1574.78 | −6.38 | −0.371 | −6.65 | −0.51 | |

Europe | 194.21 | 0.25 | 0.159 | 0.28 | 0.21 | |

Oceania | 44.29 | −0.33 | −0.304 | −0.07 | −0.27 | |

Africa | 633.57 | −0.04 | −0.112 | −0.13 | −0.11 | |

Asia | 316.13 | −0.40 | −0.151 | −0.36 | −0.25 | |

India | 420.07 | −2.30 | −0.704 | −2.40 | −0.46 |

**Table 5.**Global-averaged results of annual-mean and annual-maximum 1-day streamflow trend analysis—WBM streamflow simulations based on the WFD observation-based inputs as well as from the ISI-MIP (WBM model driven by GCM climate forcing), for 1971–2001. The 5 ISI-MIP models give 5 global averages, of which the minimum, maximum, median, mean, and standard deviation are presented. The average discharge magnitude (Q) values are per grid cell. Ave. = Average, Min. = Minimum, Max. = Maximum, Med. = Median, St. Dev. = Standard Deviation.

Discharge Ave (Q) (m^{3}·s^{−1}) | Slope of Change (b) (m^{3}·s^{−1}·year^{−1}) | Relative Change (b/Q) (%·year^{−1}) | Qmed (m^{3}·s^{−1}·year^{−1}) | Z Score (-) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean Disch. | Max. 1d Disch. | Mean Disch. | Max. 1d Disch. | Mean Disch. | Max. 1d Disch. | Mean Disch. | Max. 1d Disch. | Mean Disch. | Max. 1d Disch. | |

WFD | 116.50 | - | −0.28 | - | −0.041 | - | −0.29 | - | −0.06 | - |

ISI-MIP Ave. | 132.0 | 570.3 | −0.02 | 0.41 | 0.028 | 0.032 | −0.04 | 0.19 | 0.01 | 0.02 |

ISI-MIP Min | 128.3 | 531.6 | −0.21 | −0.46 | −0.015 | −0.015 | −0.21 | −0.52 | −0.01 | 0.00 |

ISI-MIP Max | 138.7 | 620.9 | 0.14 | 1.13 | 0.059 | 0.058 | 0.15 | 0.59 | 0.03 | 0.04 |

ISI-MIP Med. | 130.6 | 568.2 | 0.02 | 0.66 | 0.033 | 0.031 | −0.02 | 0.31 | 0.00 | 0.01 |

ISI-MIP St. Dev. | 4.0 | 33.1 | 0.15 | 0.64 | 0.027 | 0.029 | 0.14 | 0.43 | 0.02 | 0.02 |

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**MDPI and ACS Style**

Asadieh, B.; Krakauer, N.Y.; Fekete, B.M. Historical Trends in Mean and Extreme Runoff and Streamflow Based on Observations and Climate Models. *Water* **2016**, *8*, 189.
https://doi.org/10.3390/w8050189

**AMA Style**

Asadieh B, Krakauer NY, Fekete BM. Historical Trends in Mean and Extreme Runoff and Streamflow Based on Observations and Climate Models. *Water*. 2016; 8(5):189.
https://doi.org/10.3390/w8050189

**Chicago/Turabian Style**

Asadieh, Behzad, Nir Y. Krakauer, and Balázs M. Fekete. 2016. "Historical Trends in Mean and Extreme Runoff and Streamflow Based on Observations and Climate Models" *Water* 8, no. 5: 189.
https://doi.org/10.3390/w8050189