# Monthly Rainfall-Runoff Modeling at Watershed Scale: A Comparative Study of Data-Driven and Theory-Driven Approaches

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## Abstract

**:**

## 1. Introduction

## 2. Study Watersheds and Data Sources

## 3. Methodology

#### 3.1. Theory-Driven Conceptual Hydrologic Model

_{d}is surface runoff (mm), P is precipitation (mm), λ is initial wetting fraction, and W is wetting capacity (mm). In this model, $P$ is the only monthly input, while $\lambda $ and $W$ are parameters with fixed values for each watershed. In addition, as part of model input preparation, we divided monthly data into energy-limited months with subscript e and water-limited months with subscript w, based on the mean monthly aridity index [25]:

_{e}, λ

_{w}, W

_{e}, and W

_{w}were calibrated using the data from 1983 to 1992. The calibration was done by simulating surface runoff using all possible combinations of parameter values within predefined parameter value ranges [26] and selecting the combination with the highest Nash–Sutcliffe efficiency (NSE) value. The calibrated PHM was then used to simulate monthly surface runoff using precipitation as input in the period 1993 to 2002 for model validation.

#### 3.2. Data-Driven Method

_{d}:

#### 3.3. Comparative Study, Sensitivity Analysis, and Uncertainty Analysis

## 4. Results

#### 4.1. Model Performance of PHM and GPR

#### 4.2. Sensitivity Analysis

#### 4.3. Uncertainty Analysis

## 5. Discussion

#### 5.1. Model Performance Comparison

#### 5.2. Physical Interpretation of Sensitivity Analysis Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The 203 study watersheds selected for this study from the Model Parameter Estimation Experiment (MOPEX) database with a wide range of mean annual aridity index.

**Figure 2.**Exceedance frequency curves of Nash–Sutcliffe efficiency (NSE) values of proportionality hydrologic model (PHM), Gaussian process regression (GPR), and extended Gaussian process regression (EGPR) simulations.

**Figure 3.**Spatial distribution of NSE values in MOPEX in the continental US: (

**a**) PHM, (

**b**) GPR, and (

**c**) EGPR.

**Figure 4.**Comparison of runoff simulation results between PHM, GPR, and EGPR in time series. (

**a**) Wenatchee River, WA (Gauge ID 12459000); (

**b**) Yellowstone River, UT (Gauge ID 9292500); (

**c**) Little Blue River, KS (Gauge ID 6884400); (

**d**) East Fork White River, IN (Gauge ID 3365500).

**Figure 5.**PHM simulated surface runoff sensitivity to precipitation change in study watersheds with positive NSE values in (

**a**) water-limited season and (

**b**) energy-limited season. In order to show the main trend, median curves are highlighted as black lines.

**Figure 6.**GPR sensitivity to change in each input variable of all 203 watersheds in (

**a**,

**c**) water-limited season and (

**b**,

**d**) energy-limited season. Different colors show the cluster memberships of individual curves.

**Figure 7.**EGPR sensitivity to change in each input variable of all 203 watersheds in (

**a**,

**c**,

**e**,

**g**) water-limited season, and (

**b**,

**d**,

**f**,

**h**) energy-limited season. Different colors show the cluster memberships of individual curves.

**Figure 8.**Spatial distribution of watershed clustering in GPR, including surface runoff (Q

_{d}) sensitivity to precipitation (P) in (

**a**) water-limited seasons and (

**b**) energy-limited seasons; and surface runoff sensitivity to mean monthly aridity index (AI) in (

**c**) water-limited seasons and (

**d**) energy-limited seasons.

**Figure 9.**Spatial distribution of watershed clustering in EGPR, including surface runoff sensitivity to (

**a**) precipitation, (

**c**) mean monthly aridity index, (

**e**) potential evaporation (E

_{p}) and (

**g**) NDVI in water-limited seasons and (

**b**,

**d**,

**f**,

**h**) in energy-limited seasons.

**Figure 10.**Prediction intervals for GPR (left panels) and EGPR (right panels) for (

**a**,

**b**) Wenatchee River, WA (Gauge ID 12459000); (

**c**,

**d**) Yellowstone River, UT (Gauge ID 9292500); (

**e**,

**f**) Little Blue River, KS (Gauge ID 6884400); and (

**g**,

**h**) East Fork White River, IN (Gauge ID 3365500).

**Figure 11.**Box plots for catchment-wise (

**a**) actual coverage and (

**b**) width of 95% prediction interval for all watersheds with NSE > 0.

**Figure 12.**Correlation coefficient between surface runoff observation and input variables of (

**a**) precipitation, (

**b**) aridity index, (

**c**) potential evaporation, and (

**d**) NDVI.

**Figure 13.**Additional GPR using precipitation and potential evaporation as input variables. Sensitivity to changes of potential evaporation in all 203 watersheds in (

**a**) water-limited seasons and (

**b**) energy-limited seasons. Different colors show the cluster memberships of individual curves.

**Table 1.**Number of watersheds in each quantile of NSE and normalized root mean square error (NRMSE) values and mean NSE and NRMSE values of PHM, GPR, and EGPR.

Model | NSE | NRMSE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

<0 | 0–0.5 | 0.5–0.9 | 0.9–1 | Mean | 0–0.5 | 0.5–1 | 1–2 | >2 | Mean | |

PHM | 30 | 79 | 94 | 0 | 0.38 | 0 | 94 | 103 | 6 | 1.14 |

GPR | 19 | 84 | 100 | 0 | 0.39 | 1 | 100 | 94 | 8 | 1.15 |

EGPR | 6 | 57 | 139 | 1 | 0.52 | 3 | 124 | 72 | 4 | 1.01 |

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**MDPI and ACS Style**

Chang, W.; Chen, X. Monthly Rainfall-Runoff Modeling at Watershed Scale: A Comparative Study of Data-Driven and Theory-Driven Approaches. *Water* **2018**, *10*, 1116.
https://doi.org/10.3390/w10091116

**AMA Style**

Chang W, Chen X. Monthly Rainfall-Runoff Modeling at Watershed Scale: A Comparative Study of Data-Driven and Theory-Driven Approaches. *Water*. 2018; 10(9):1116.
https://doi.org/10.3390/w10091116

**Chicago/Turabian Style**

Chang, Won, and Xi Chen. 2018. "Monthly Rainfall-Runoff Modeling at Watershed Scale: A Comparative Study of Data-Driven and Theory-Driven Approaches" *Water* 10, no. 9: 1116.
https://doi.org/10.3390/w10091116