# Comparison of the Calibrated Objective Functions for Low Flow Simulation in a Semi-Arid Catchment

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Study Area

#### 2.2. Data

## 3. Methods

#### 3.1. Hydrological Model and Model Optimization

#### 3.2. Calibration Objective Functions

Classes | Criteria | Name | Description | Reference |
---|---|---|---|---|

Single objective | KGE_{(log(Q))} | OBJ1 | KGE calculated on logarithmic transformed discharges | Oudin et al. [33] |

KGE_{(1/Q)} | OBJ2 | KGE calculated on inverse transformed discharges | Pushpalatha et al. [34] | |

Muti objective | KGE_{(Q)}+KGE_{(log(Q))} | OBJ3 | Sum of KGE calculated on discharges and logarithmic transformed discharges | Proposed in this study |

KGE_{(Q)}+KGE_{(1/Q)} | OBJ4 | Sum of KGE calculated on discharges and inverse transformed discharges | Garcia et al. [3] | |

KGE_{(Qsort)}+KGE_{(log(Qsort))} | OBJ5 | Sum of KGE calculated on the FDC and logarithmic transformed of the FDC | Proposed in this study | |

KGE_{(Qsort)}+KGE_{(1/Qsort)} | OBJ6 | Sum of KGE calculated on the FDC and logarithmic transformed of the FDC | Garcia et al. [3] | |

Split objective | split KGE_{(Q)} | OBJ7 | Averaged KGE calculated on discharges in each year | Fowler et al. [35] |

split (KGE_{(Q)}+KGE_{(1/Q)}) | OBJ8 | Averaged sum of KGE calculated on discharges and inverse transformed discharges in each year | Proposed in this study |

#### 3.3. Model Performance Assessment

#### Climatic Robustness Assessment

#### 3.4. Assessment Criteria

## 4. Results

#### 4.1. Objective Functions Evaluation

#### 4.1.1. Hydrograph Simulation

_{log}) are also included. Table 3 presents the calculated values of KGE and KGE

_{log}during two calibration periods with all eight objective functions. In the table, the highest values for each period among objective functions are highlighted in bold, and ‘/’ is used when the value is lower than 0.

_{log}, three objectives produce values lower than 0, which means unacceptable. At the same time, all the multi objective functions provide good results, whose KGE

_{log}values are higher than 0.61. The highest KGE

_{log}value appears for OBJ1; this is mainly because the evaluation criterion is the same as the objective function and the KGE

_{log}values for OBJ3 are very close to OBJ1.

_{log}values for the two periods as the example, the result is 0.821 and 0.797 for OBJ3 and OBJ4, respectively, followed by 0.773 for OBJ1. Among the multi objectives, regardless of whether it is time series-based or FDC-based, the logarithmic transformed objectives tend to yield higher averaged measurements than the inverse transformed objectives. The averaged KGE and KGE

_{log}value of the two periods for OBJ5 is 0.736, which is 0.685 for OBJ6.

#### 4.1.2. Flow Duration Curves

#### 4.1.3. Low Flow Indices

^{3}/s from the inverse transformed objectives, which is about 2.2 m

^{3}/s from the logarithmic transformed objectives.

^{3}/s from the inverse transformed objectives, which climbs to 0.8 m

^{3}/s from the logarithmic partners. Conversely, the logarithmic transformed objectives provide a better estimation for the indices less sensitive to the extreme low flows (e.g., MAM30 and Q75). Observing the subplot for MAM30, the averaged estimation error is about 1.4 m

^{3}/s from the inverse transformed objectives, which is about 13 times for the logarithmic partners.

#### 4.2. Climatic Robustness Assessment

#### 4.2.1. Hydrograph Simulation

_{log}) is presented in Table 4 to provide more valuable information for hydrograph simulation evaluation.

#### 4.2.2. Flow Duration Curves

#### 4.2.3. Low Flow Indices

## 5. Discussion

#### 5.1. Objective Functions Evaluation

#### 5.2. Climatic Robustness Assessment

## 6. Conclusions

- -
- The influence of the included transformation formats in objective functions on low flow simulation is pronounced, and logarithmic transformation is recommended.
- -
- Among the three classes of objective functions, the combined multi-class is highly recommended, and the mean of KGE
_{(Q)}and KGE_{(log(Q))}remains a first choice. In contrast, the class of split objectives is regarded as the last choice as it demonstrated the worst performance. - -
- Replacing the objective function from the time series based on the FDC could not improve the simulation performance.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The precipitation (P) information in Ma Du Wang station, the blue and red window marks the calibration periods, and the yellow window marks the validation period.

**Figure 3.**The probability density function (PDF) comparison for the objective functions evaluating by the percent bias (Pbias) during the calibration period 2003–2005.

**Figure 4.**The probability density function (PDF) comparison for the objective functions evaluating by the percent bias (Pbias) during the calibration period 2007–2009.

**Figure 5.**The result of the observed and simulated FDCs by all objective functions during the calibration period 2003–2005. The zoomed plots in each subplot show the result for the highest 10% flow (

**left**) and the lowest 50% flow (

**right**) simulations.

**Figure 6.**The result of the observed and simulated FDCs by all objective functions during the calibration period 2007–2009. The zoomed plots in each subplot show the result for the highest 10% flow (

**left**) and the lowest 50% flow (

**right**) simulations.

**Figure 7.**The observed (the line) and simulated (the bars) low flow indices by all objective functions during the calibration period 2003–2005.

**Figure 8.**The observed (the line) and simulated (the bars) low flow indices by all objective functions during the calibration period 2007–2009.

**Figure 9.**The hydrograph plot during the validation period based on the calibration period (

**a**) 2003–2005 (

**b**) 2007–2009.

**Figure 10.**The observed and simulated FDCs for the validation period based on the calibration in (

**a**) 2003–2005 (

**b**) 2007–2009. The zoomed plots in each subplot show the result for the highest 10% flow (

**left**) and the lowest 50% flow (

**right**) simulations.

**Figure 11.**The observed and simulated low flow indices by all objective functions during the validation period.

Criteria | Description |
---|---|

KGE | Kling-Gupta Efficiency (see Equation (1)) |

KGE_{log} | KGE calculated on logarithmic transformed flow |

MAM3 | Mean Annual Minimum 3-day mean flow at 3-year return period |

MAM10 | Mean Annual Minimum 10-day mean flow at 3-year return period |

MAM30 | Mean Annual Minimum 30-day mean flow at 3-year return period |

LFD | The duration of low flow smaller than 30% of the time |

Q95 | Flow exceeded 95% of the time |

Q75 | Flow exceeded 75% of the time |

Evaluation Criteria | KGE | KGElog | ||
---|---|---|---|---|

Calibration Period | 2003–2005 | 2007–2009 | 2003–2005 | 2007–2009 |

OBJ1 | 0.85 | 0.63 | 0.78 | 0.84 |

OBJ2 | 0.60 | 0.25 | / | / |

OBJ3 | 0.90 | 0.78 | 0.77 | 0.83 |

OBJ4 | 0.92 | 0.78 | 0.70 | 0.79 |

OBJ5 | 0.89 | 0.62 | 0.74 | 0.69 |

OBJ6 | 0.90 | 0.55 | 0.68 | 0.61 |

OBJ7 | 0.85 | 0.68 | / | / |

OBJ8 | 0.74 | 0.69 | / | / |

Calibration Period | 2003–2005 | 2007–2009 | ||
---|---|---|---|---|

Evaluation Criteria | KGE | KGElog | KGE | KGElog |

OBJ1 | 0.61 | 0.67 | 0.42 | 0.69 |

OBJ3 | 0.68 | 0.70 | 0.58 | 0.68 |

OBJ4 | 0.68 | 0.62 | 0.61 | 0.71 |

OBJ5 | 0.79 | 0.67 | 0.58 | 0.63 |

OBJ6 | 0.61 | 0.64 | 0.49 | 0.67 |

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**MDPI and ACS Style**

Yang, X.; Yu, C.; Li, X.; Luo, J.; Xie, J.; Zhou, B. Comparison of the Calibrated Objective Functions for Low Flow Simulation in a Semi-Arid Catchment. *Water* **2022**, *14*, 2591.
https://doi.org/10.3390/w14172591

**AMA Style**

Yang X, Yu C, Li X, Luo J, Xie J, Zhou B. Comparison of the Calibrated Objective Functions for Low Flow Simulation in a Semi-Arid Catchment. *Water*. 2022; 14(17):2591.
https://doi.org/10.3390/w14172591

**Chicago/Turabian Style**

Yang, Xue, Chengxi Yu, Xiaoli Li, Jungang Luo, Jiancang Xie, and Bin Zhou. 2022. "Comparison of the Calibrated Objective Functions for Low Flow Simulation in a Semi-Arid Catchment" *Water* 14, no. 17: 2591.
https://doi.org/10.3390/w14172591