# Development of a One-Parameter New Exponential (ONE) Model for Simulating Rainfall-Runoff and Comparison with Data-Driven LSTM Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Hydrological Data

^{2}, the total water storage is 2900 million m

^{3}and the effective water storage capacity is 1900 million m

^{3}. The annual water supply is 1213 million m

^{3}with a flood control capacity of 500 million m

^{3}. The Yongdam (YD) Dam, constructed in 2001, is a multipurpose dam located upstream of the Geum River Basin that supplies domestic, industrial and agricultural water to six regions in Jeollabuk-do and two regions in the Chungcheong Province. It is a concrete-faced rockfill dam with a height and length of 70 m and 498 m, respectively. The watershed area is 930 km

^{2}, the total water storage is 815 million m

^{3}and the effective water storage capacity is 672.5 million m

^{3}. The annual water supply is 1143.2 million m

^{3}, with a flood control capacity of 137 million m

^{3}.

#### 2.2. ONE Model Development

#### 2.3. LSTM Model

_{t}is the input value at the present time, h

_{t−}

_{1}is the hidden value of the past time, W is the weight, b is the bias, C

_{t}is the cell state and $\stackrel{~}{{C}_{t}}$ is a new candidate value. Second, the input gate (i) is used for memorizing the current information and is expressed by applying a sigmoid as an activation function. The expression is as follows:

#### 2.4. Performance Evaluation of the Runoff Model

^{2}), root mean square error (RMSE) and percent bias (PBIAS). NSE and R

^{2}have a range value and show that the closer their values are to 1, the better the predictive ability of the model is. RMSE is an index that indicates the difference between the simulated and observed values and ranges from 0 to ∞; a value closer to 0 indicates a higher simulation accuracy of the model. It is possible to evaluate the concentration of the simulated values on the best-fitting line. PBIAS represents the average tendency between the observed and simulated values and ranges from 0 to ∞, with values closer to 0 indicating a higher simulation accuracy of the model [49,50,51,52,53].

## 3. Results

#### 3.1. Parameter Sensitivity of the ONE Model

^{2}, RMSE and PBIAS values were plotted at 0.1 intervals. The NSE values of the SY and YD dams ranged from 0.1 to 0.9 and the R

^{2}values ranged from 0.79 to 0.97. Both NSE and R

^{2}reached their maximum values near parameter 2, starting from a positive value and then appearing as a concave curve. The RMSE index starts from a positive value of 6 mm/day, has a minimum value when the parameter value reaches 2 and assumes a convex curve that increases again with the increase in the value of the parameter. This index shows a continuously increasing trend with only positive values. For the PBIAS index, a positive value of approximately 60% shows a negative value as the parameter increases. Although this index exhibits a continuous tendency to decrease, the slope of the evaluation index decreases as the parameter increases.

^{2}both exhibited concave curves with positive values and the maximum value can be selected as an appropriate parameter. As the slope of the curve for the NSE index is large, the appropriate parameter with the maximum value can be clearly identified. Because the PBIAS evaluation index can continuously decrease from a positive to a negative value, it is appropriate to select an optimal parameter so that the difference between observed and simulated value is 0 with the smallest error. Furthermore, because the RMSE evaluation index increases as the observed simulation error decreases, selecting an appropriate parameter may result in a parameter with a minimum value.

^{2}evaluation indices were above 0.7. In addition, the maximum value of NSE where the curve of the evaluation index clearly appears can be confirmed. The range of the minimum value of the RMSE can be referred to and the PBIAS evaluation index can be determined as a parameter with a value close to 0. The ONE model is used for long-term runoff simulation and the PBIAS evaluation index is important owing to the total volume error of runoff over the entire period. Therefore, the user can determine the parameter of the corresponding point by checking the parameter (w) with a PBIAS index close to 0 and comprehensively review the results of the remaining NSE, R

^{2}and RMSE values.

#### 3.2. Comparison of Runoff Simulation Results Obtained by ONE and LSTM

^{2}, RMSE and PBIAS. Table 2 presents the results of the statistical analysis of the daily runoff of the ONE and LSTM models. Each calibration (training) and test period was 2002–2015 and 2016–2021, respectively. The ONE model has one parameter each for the SY Dam and YD Dam, whereas the parameters of the LSTM model are the results calculated using 32,104 and 24,076 parameters for the SY Dam and YD Dam, respectively.

^{2}, 80.67 m

^{3}/s for RMSE and −1.07% for PBIAS during the calibration period. For the validation period, each evaluation index was calculated as 0.86, 0.93, 74.38 m

^{3}/s and 0.08% and the simulation results of the ONE model well reflected the observed values. The runoff of the LSTM model was calculated as 0.69 for NSE, 0.83 for R

^{2}, 125.72 m

^{3}/s for RMSE and 8.86% for PBIAS over the training period. For the validation period, each evaluation index was calculated as 0.58, 0.76, 128.04 m

^{3}/s and 8.82%. For the YD Dam, the runoff of the ONE model was calculated as 0.87 for NSE, 0.93 for R

^{2}, 27.81 m

^{3}/s for RMSE and 2.35% for PBIAS over the calibration period. For the validation period, each evaluation index was calculated as 0.91, 0.95, 25.10 m

^{3}/s and 9.95% and the simulation results of the ONE model accurately reflected the observed values. The runoff of the LSTM model was calculated as 0.52 for NSE, 0.72 for R

^{2}, 53.37 m

^{3}/s for RMSE and −0.37% for PBIAS over the training, respectively. For the validation period, the evaluation indices were calculated as 0.44, 0.67, 62.73 m

^{3}/s and 1.58%.

^{2}index, the ONE model exhibited a value of 0.9 or higher, which is evaluated as “very good” depending on the criteria presented in the study [56]. In the LSTM model, the SY Dam and YD Dam were evaluated as “very good” and “good” at 0.82 and 0.70, respectively. In terms of the RMSE index, the ONE model was 47.59 m

^{3}/s and 29.30 m

^{3}/s lower than that of the LSTM model for the SY Dam and YD Dam, respectively, which decreases the error. In terms of the PBIAS index, both the ONE and LSTM models had an absolute error of less than 10% and were evaluated as “very good” according to the standard suggested by Van Liew et al. [57]. Among all four evaluation indices, the ONE model showed a better runoff simulation performance.

#### 3.3. Comparison of Monthly and Annual Runoff Rate Calculation Results

^{2}values of the ONE and LSTM models were 0.89 and 0.48, respectively. In the case of the YD Dam, the R

^{2}values of the ONE and LSTM models were 0.79 and 0.66, respectively. The annual runoff rates predicted by the ONE and LSTM models were evaluated as “very good” (0.75 < R

^{2}< 1.00) and “good” (0.6 < R

^{2}< 0.75) or “bad” (0.25 < R

^{2}< 0.50), respectively, according to the evaluation classification used by Fernandez et al. [56]. When plotting the scatter plot of the runoff rate, the original model was concentrated on a 1:1 line compared to that of the LSTM, thereby confirming that the distribution of the YD Dam was wider than that of the SY Dam.

## 4. Discussion

## 5. Conclusions

^{2}, RMSE and PBIAS. Furthermore, the runoff simulation performance of the ONE model was superior to that of the learning-based LSTM model in terms of the daily simulation values, monthly heatmaps and annual runoff rates. This study, therefore, demonstrated the possibility of simulating rainfall-runoff using only one parameter and highlights that the practical utility of a hydrological model can be improved by reducing the number of parameters. The use of a single parameter is expected to maximize user convenience for simulating runoff, which is essential in the operation of water resource facilities. In the future, the ONE model will be reviewed for applicability to multipurpose dams across Korea. In addition, a regression equation using data for the rainfall and runoff rate of multipurpose dams will be derived and used as a referential index to determine one parameter of the ONE model. This will expand the model for use in agricultural reservoirs with no inflow data.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Changes in the performance evaluation indices: (

**a**) NSE, (

**b**) R

^{2}, (

**c**) RMSE and (

**d**) PBIAS according to the parameter value used by the ONE model at the SY Dam.

**Figure 6.**Changes in the performance evaluation indices: (

**a**) NSE, (

**b**) R

^{2}, (

**c**) RMSE and (

**d**) PBIAS according to the parameter value used by the ONE model for the YD Dam.

**Figure 7.**Comparison of the runoff simulation observation results obtained by the ONE and LSTM models at the SY Dam.

**Figure 8.**Comparison of the runoff simulation observation results obtained by the ONE and LSTM models at the YD Dam.

**Figure 9.**Monthly heatmap of (

**a**) the observed runoff results in comparison to that simulated by (

**b**) ONE and (

**c**) LSTM models at the SY Dam. The white frame shows major difference of monthly runoff by models.

**Figure 10.**Monthly heatmap of (

**a**) the observed runoff result in comparison to that simulated by (

**b**) ONE and (

**c**) LSTM models at the YD Dam. The white frame shows major difference of monthly runoff by models.

**Figure 11.**Scatter plot of the annual observed and simulated runoff rates at (

**a**) the SY Dam and (

**b**) the YD dam, as predicted by the ONE and LSTM models.

Dam | Watershed Area (km ^{2}) | Height/Length (m) | Total Water Storage (Million m^{3}) | Effective Storage Capacity (Million m^{3}) | Flood Control (Million m^{3}) | Water Supply (Million m^{3}) |
---|---|---|---|---|---|---|

Soyang | 2703 | 123/530 | 2900 | 1900 | 500 | 1213 |

Yongdam | 930 | 70/498 | 815 | 672.5 | 137 | 1143.2 |

Dam | Model | Number of Parameters | Period | Statistics | |||
---|---|---|---|---|---|---|---|

NSE | R^{2} | RMSE | PBIAS | ||||

Soyang | ONE | 1 | Calibration (2002–2015) | 0.87 | 0.93 | 80.67 | −1.07 |

Validation (2016–2021) | 0.86 | 0.93 | 74.38 | 0.08 | |||

Total period | 0.87 | 0.93 | 78.83 | −0.75 | |||

LSTM | 32,104 | Training (2002–2015) | 0.69 | 0.83 | 125.72 | 8.86 | |

Validation (2016–2021) | 0.58 | 0.76 | 128.04 | 8.82 | |||

Total period | 0.67 | 0.82 | 126.42 | 8.68 | |||

Yongdam | ONE | 1 | Calibration (2002–2015) | 0.87 | 0.93 | 27.81 | 2.35 |

Validation (2016–2021) | 0.91 | 0.95 | 25.10 | 9.95 | |||

Total period | 0.88 | 0.94 | 27.03 | 4.57 | |||

LSTM | 24,076 | Training (2002–2015) | 0.52 | 0.72 | 53.37 | −0.37 | |

Validation (2016–2021) | 0.44 | 0.67 | 62.73 | 1.58 | |||

Total period | 0.49 | 0.70 | 56.33 | 0.20 |

Dam | Model | Rainfall (mm) | Runoff (mm) | Runoff Ratio (%) |
---|---|---|---|---|

Soyang | ONE | 1221.3 | 723.1 | 59.2 |

LSTM | 780.3 | 63.9 | ||

Observed | 722.5 | 59.2 | ||

Yongdam | ONE | 1429.8 | 879.9 | 61.5 |

LSTM | 811.6 | 56.8 | ||

Observed | 800.3 | 56.0 |

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**MDPI and ACS Style**

Lee, J.; Noh, J.
Development of a One-Parameter New Exponential (ONE) Model for Simulating Rainfall-Runoff and Comparison with Data-Driven LSTM Model. *Water* **2023**, *15*, 1036.
https://doi.org/10.3390/w15061036

**AMA Style**

Lee J, Noh J.
Development of a One-Parameter New Exponential (ONE) Model for Simulating Rainfall-Runoff and Comparison with Data-Driven LSTM Model. *Water*. 2023; 15(6):1036.
https://doi.org/10.3390/w15061036

**Chicago/Turabian Style**

Lee, Jaenam, and Jaekyoung Noh.
2023. "Development of a One-Parameter New Exponential (ONE) Model for Simulating Rainfall-Runoff and Comparison with Data-Driven LSTM Model" *Water* 15, no. 6: 1036.
https://doi.org/10.3390/w15061036