# Parameter Estimation for Soil Water Retention Curve Using the Salp Swarm Algorithm

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## Abstract

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## 1. Introduction

## 2. The Van Genuchten Model

## 3. Salp Swarm Algorithm

Algorithm 1 The Procedure of the Salp Swarm Algorithm (SSA) Algorithm. |

Require: Initialize the salp population ${x}_{i}(i=1,2,...,n)$ consider $ub$ and $lb$.while (End condition is not satisfied) Calculate the fitness of each search salp F=the best search solution Update ${r}_{1}$ by Equation (4) for each salp (${x}_{i}$)if ($i==1$) Update the position of the leading salp by Equation (3) esle Update the position of the followers salp by Equation (6) endend Verify the position of salps based on the upper and lower bounds endreturnF |

#### Benchmarking Algorithms

## 4. Estimation Algorithms and Dataset

#### Data Description

## 5. Estimation Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The estimated SWRCs of eight soil samples using SSA. (

**a**) Sample 3020, (

**b**) Sample 1120, (

**c**) Sample 1102, (

**d**) Sample 1330, (

**e**) Sample 1162, (

**f**) Sample 2400, (

**g**) Sample 1361, and (

**h**) Sample 1173.

Soil Sample ID | Location | Bulk Density | Data Number | Soil Type |
---|---|---|---|---|

3020 | Moscow, Russia | 1.21 | 5 | Sand |

1120 | Rome, AL, USA | 1.63 | 10 | Sandy Loam |

3154 | Dickey Co., ND, USA | 1.53 | 10 | Sand |

1330 | Hannover, Germany | 1.37 | 21 | Silt |

1173 | Clemson, SC, USA | 1.38 | 11 | Clay Loam |

1102 | Blackville, SC, USA | 1.71 | 9 | Sandy Clay |

1162 | Watkinsville, GA, USA | 1.54 | 15 | Clay |

1361 | Reinhausen (Goettingen), Germany | 1.49 | 11 | Silty Clay |

2400 | Cass County, ND, USA | 1.08 | 17 | Loam |

Parameter | ${\mathit{\phi}}_{\mathit{r}}$ | ${\mathit{\phi}}_{\mathit{s}}$ | $\mathit{\alpha}$ | n |
---|---|---|---|---|

Lower Bound | 0 | 0 | 0 | 1 |

Upper Bound | 1 | 1 | 100 | 100 |

**Table 3.**Estimated parameters and the sum of squared errors (SSE) of the van Genuchten model using SSA, differential evolution (DE), particle swarm optimization (PSO), and RETC.

Soil Sample ID | Algorithm | ${\mathit{\phi}}_{\mathit{r}}$ (cm${}^{3}$cm${}^{-3}$) | ${\mathit{\phi}}_{\mathit{s}}$ (cm${}^{3}$cm${}^{-3}$) | $\mathit{\alpha}$ (cm${}^{-1}$) | n | $\mathbf{SSE}\left({10}^{-3}\right)$ |
---|---|---|---|---|---|---|

3020 | SSA | 0.19166 | 0.44901 | 0.06244 | 2.40869 | 0.011171 |

DE | 0.19166 | 0.44901 | 0.01796 | 2.40869 | 0.011724 | |

RETC | 0.19166 | 0.44901 | 0.01796 | 2.40885 | 0.011718 | |

PSO | 0.19220 | 0.44900 | 0.01780 | 2.43342 | 0.011732 | |

1120 | SSA | 0.07951 | 0.28940 | 0.02583 | 1.88130 | 0.261462 |

DE | 0.07951 | 0.28940 | 0.01238 | 1.88130 | 0.261464 | |

RETC | 0.07951 | 0.28940 | 0.01239 | 1.88119 | 0.261465 | |

PSO | 0.07951 | 0.28940 | 0.01238 | 1.88131 | 0.261463 | |

3154 | SSA | 0.06941 | 0.41611 | 0.06706 | 2.69126 | 0.214173 |

DE | 0.06941 | 0.41611 | 0.02813 | 2.69128 | 0.214174 | |

RETC | 0.06942 | 0.41609 | 0.02813 | 2.69178 | 0.214175 | |

PSO | 0.06941 | 0.41611 | 0.02813 | 2.69131 | 0.214174 | |

1330 | SSA | 0.08362 | 0.38004 | 0.00337 | 2.11588 | 11.25378 |

DE | 0.08362 | 0.38004 | 0.00259 | 2.11588 | 11.25378 | |

RETC | 0.08373 | 0.37998 | 0.00259 | 2.11992 | 11.25386 | |

PSO | 0.08344 | 0.38021 | 0.00260 | 2.10977 | 11.25391 | |

1173 | SSA | 0.29087 | 0.47857 | 0.03335 | 1.14316 | 0.037139 |

DE | 0.30479 | 0.47850 | 0.03022 | 1.16004 | 0.037951 | |

RETC | 0.30130 | 0.47850 | 0.04960 | 1.15530 | 0.499869 | |

PSO | 0.03266 | 0.47909 | 0.05867 | 1.04872 | 0.041419 | |

1102 | SSA | 0.12175 | 0.34674 | 0.09208 | 1.29730 | 0.240458 |

DE | 0.12198 | 0.34673 | 0.09168 | 1.29806 | 0.240460 | |

RETC | 0.12170 | 0.34670 | 0.15910 | 1.29720 | 1.402827 | |

PSO | 0.12183 | 0.34677 | 0.09208 | 1.29753 | 0.240460 | |

1162 | SSA | 0.29374 | 0.41333 | 0.01369 | 1.31096 | 2.711846 |

DE | 0.29372 | 0.41334 | 0.01371 | 1.31075 | 2.711846 | |

RETC | 0.29400 | 0.41330 | 0.03770 | 1.31270 | 4.622924 | |

PSO | 0.29311 | 0.41346 | 0.01424 | 1.30620 | 2.711884 | |

1361 | SSA | 0.16454 | 0.43307 | 0.00105 | 1.25767 | 0.266938 |

DE | 0.16448 | 0.43308 | 0.00106 | 1.25746 | 0.266939 | |

RETC | 0.16450 | 0.43319 | 0.00430 | 1.25760 | 10.92369 | |

PSO | 0.16128 | 0.43329 | 0.00112 | 1.25138 | 0.267039 | |

2400 | SSA | 0.19672 | 0.45414 | 0.00139 | 1.58303 | 0.144518 |

DE | 0.19633 | 0.45419 | 0.00139 | 1.58259 | 0.144576 | |

RETC | 0.19670 | 0.45410 | 0.00157 | 1.58310 | 0.418468 | |

PSO | 0.19455 | 0.45444 | 0.00147 | 1.57232 | 0.144657 |

Soil Sample Id | SSE on Raspberry Pi 3 $\left({10}^{-3}\right)$ | SSE on Windows 10 System $\left({10}^{-3}\right)$ |
---|---|---|

2400 | 0.144518 | 0.144518 |

3020 | 0.011171 | 0.011171 |

1361 | 0.266938 | 0.266938 |

1102 | 0.240458 | 0.240458 |

1330 | 11.25378 | 11.25378 |

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**MDPI and ACS Style**

Zhang, J.; Wang, Z.; Luo, X.
Parameter Estimation for Soil Water Retention Curve Using the Salp Swarm Algorithm. *Water* **2018**, *10*, 815.
https://doi.org/10.3390/w10060815

**AMA Style**

Zhang J, Wang Z, Luo X.
Parameter Estimation for Soil Water Retention Curve Using the Salp Swarm Algorithm. *Water*. 2018; 10(6):815.
https://doi.org/10.3390/w10060815

**Chicago/Turabian Style**

Zhang, Jing, Zhenhua Wang, and Xiong Luo.
2018. "Parameter Estimation for Soil Water Retention Curve Using the Salp Swarm Algorithm" *Water* 10, no. 6: 815.
https://doi.org/10.3390/w10060815