# Differential Evolution’s Application to Estimation of Soil Water Retention Parameters

## Abstract

**:**

## 1. Introduction

## 2. Differential Evolution

#### 2.1. Initialization

#### 2.2. Mutation

#### 2.3. Crossover

#### 2.4. Selection

## 3. Fitting Soil Water Retention Parameters to Hydraulic Data

- The choice of dataset. There are four tables about preshead-θ data in UNSODA: field_drying_h-t, field_wetting_h-t, lab_drying_h-t and lab_wetting_h-t. There are 127 samples in field_drying_h-t table, 0 samples in field_wetting_h-t table, 700 samples in lab_drying_h-t table, and 28 samples in lab_wetting_h-t table with available data and this dataset is called dataset 1. The data in dataset 1 are diverse and heterogeneous (field or lab, wetting or drying), and appropriate for looking into the universal characteristics of UNSODA and the feasibility and robustness of DE. Schaap and Leij [12] and Schaap et al. [13] chose 235 codes in lab_drying_h-t table as a dataset based on the criteria: quality of the data; presence of sufficient texture data; the data from the laboratory drying (drainage) branches; eliminating samples with low bulk density values (<0.5 g/cm${}^{3}$), and it is called dataset 2 here. Moreover, the designated 235 identifier codes with available data also partially appear in other two h-t tables: 26 samples in field_drying_h-t table and 1 sample in lab_wetting_h-t, the sizes of which are not enough for statistics. Table 1 and Table 2 show: The RMSE${}_{w}$ mean of dataset 1 becomes numerically stable at MEG = 400 and dataset 2 at MEG = 200; the RMSE${}_{w}$ maximum of dataset 1 is 14 times of the mean, and only 3 times for dataset 2; the discrepancy between RMSE${}_{w}$ maximum and minimum of dataset 1 is 3 times of dataset 2; the RMSE${}_{w}$ mean of dataset 2 is 20% higher than dataset 1; the means of loop times are different only at MEG ≥10,000; the minimum loop times of dataset 2 is 4∼5 times of dataset 1. The above differences of RMSE${}_{w}$ and loop times between datasets 1 and 2 lie in: Because the dataset 2 is completely from lab_drying_h-t table and the data size of any sample in dataset 2 is not less than 6, the data quality will be higher and the statistical characteristics are more homogeneous.
- The iteration termination. DE is globally convergent, the convergence speed will become slower, RMSE${}_{w}$ will not decrease for long time after reaching certain relatively steady value, and this value can be taken as a numerical convergence value (NCV). MEG or MIT shall satisfy that every sample in dataset can reach a NCV. The loop times that a sample takes to reach its own NCV is called the actual loop times and it should be less than MEG. The NCVs of RMSE${}_{w}$ mean in Table 1 and Table 2 can be respectively taken as 0.0106∼0.0107 and 0.0121∼0.0122 while means of loop times remarkably increase with MEG. In Table 2, RMSE${}_{w}$ becomes stable at MEG = 200, but the corresponding mean loop times 197 is close to MEG and it can not ensure that some samples already get NCV when MEG and mean loop times are close. Therefore it is ideal to set MEG = 1000 or above as an iteration termination condition according to the first and the sixth columns in Table 2. If a sample reaches GOV, the iteration will terminate. Hence we utilize both MEG and GOV as the termination conditions in DE programming.
- The relationship between the loop times and the data size of a soil sample. The data size is the number of data records of a soil sample, i.e., ${N}_{w}$ in Equation (3). In Table 3, MEG = 10,000; in the minimum case of soil samples, the loop times is the minimum for all samples, and the maximum likewise; number is the count of the minimum or maximum cases; but the average data sizes, 13 and 12.23 of the minimum and the maximum cases are very close and it means that loop times are unrelated to data size; the maximum cases have much lower RMSE${}_{w}$ mean and more loop times will get better fitness. It implies that DE time consumption depends on the convergency speed rather than the data size.
- The range of control variables. The rule of thumb values for the control variables in DE is: $\text{F}\in [0.5,1.0],\phantom{\rule{4pt}{0ex}}\mathrm{Cr}\in [0.8,1.0]\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\mathrm{Np}=10\xb7\text{D}$ [33]. The designated 235 identifier codes have only 26 samples with available data in the field_drying_h-t table, which composes dataset 3. We compare dataset 2 with 235 samples and dataset 3 with 26 samples by different values of Cr, 0.3 and 0.8 in Table 4 as MEG = 10,000: In dataset 2, different Crs do not generate evidently different estimates of water retention parameters; compared with dataset 2, the estimation differences in dataset 3 can not be ignored except ${\theta}_{r}$ and RMSE${}_{w}$. It means the value of Cr is not crucial to the parametric estimation in UNSODA if the dataset is big enough and this conclusion can also be safely applied to other control variables and even the initial values of parameters.

MEG | RMSE${}_{w}$ | Loop Times | Time Consumption | ||||||
---|---|---|---|---|---|---|---|---|---|

Mean | Maximum | Minimum | Std. Deviation | Mean | Maximum | Minimum | (h:m:s) | ||

50 | 0.018 | 0.193 | 0.001 | 0.014 | 46 | 49 | 13 | 00:02:38 | |

100 | 0.012 | 0.182 | 0.001 | 0.011 | 97 | 99 | 58 | 00:04:29 | |

200 | 0.011 | 0.141 | 0.001 | 0.009 | 196 | 199 | 55 | 00:08:16 | |

300 | 0.011 | 0.141 | 0.001 | 0.009 | 286 | 299 | 55 | 00:11:56 | |

400 | 0.011 | 0.141 | 0.001 | 0.009 | 356 | 399 | 52 | 00:15:28 | |

500 | 0.011 | 0.141 | 0.001 | 0.009 | 426 | 499 | 51 | 00:19:08 | |

1000 | 0.011 | 0.141 | 0.001 | 0.009 | 748 | 999 | 50 | 00:37:11 | |

10,000 | 0.011 | 0.141 | 0.001 | 0.009 | 3322 | 9999 | 49 | 07:51:09 | |

40,000 | 0.011 | 0.141 | 0.001 | 0.009 | 10,247 | 39,999 | 57 | 27:15:55 | |

Std. deviation | $2.53\times {10}^{-3}$ | $2.07\times {10}^{-2}$ | $1.52\times {10}^{-4}$ | $1.44\times {10}^{-3}$ |

MEG | RMSE${}_{w}$ | Loop Times | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Maximum | Minimum | Std. Deviation | Mean | Maximum | Minimum | ||

50 | 0.018 | 0.065 | 0.002 | 0.010 | 47 | 49 | 25 | |

100 | 0.013 | 0.039 | 0.001 | 0.007 | 97 | 99 | 83 | |

200 | 0.012 | 0.039 | 0.001 | 0.007 | 197 | 199 | 183 | |

300 | 0.012 | 0.039 | 0.001 | 0.007 | 287 | 299 | 218 | |

400 | 0.012 | 0.039 | 0.001 | 0.007 | 353 | 399 | 221 | |

500 | 0.012 | 0.039 | 0.001 | 0.007 | 424 | 499 | 221 | |

1000 | 0.012 | 0.039 | 0.001 | 0.007 | 722 | 999 | 254 | |

10,000 | 0.012 | 0.039 | 0.001 | 0.007 | 2963 | 9999 | 240 | |

40,000 | 0.012 | 0.039 | 0.001 | 0.007 | 7838 | 39,999 | 265 | |

Std. deviation | $2.067\times {10}^{-3}$ | $9.24\times {10}^{-3}$ | $3.26\times {10}^{-4}$ | $9.84\times {10}^{-4}$ |

Case | Loop Times | Number | Data Size | RMSE${}_{w}$ Mean |
---|---|---|---|---|

minimum | 240 | 1 | 13 | 0.017 |

maximum | 9999 | 31 | 12.23 | 0.009 |

Dataset 2 | Dataset 3 | ||||
---|---|---|---|---|---|

Cr = 0.3 | Cr = 0.8 | Cr = 0.3 | Cr = 0.8 | ||

${\theta}_{r}$ | 0.054 | 0.057 | 0.166 | 0.152 | |

${\theta}_{s}$ | 0.463 | 0.466 | 0.442 | 0.476 | |

$lg\left(\alpha \right)$ | −1.625 | −1.607 | −1.581 | −1.495 | |

$lg\left(n\right)$ | 0.214 | 0.208 | 0.604 | 0.536 | |

RMSE${}_{w}$ | 0.001 | 0.001 | 0.015 | 0.015 |

**Table 5.**Average hydraulic parameters for each soil textural group with standard deviations in parentheses.

N | ${\theta}_{r}$ (cm${}^{3}$cm${}^{-3}$) | ${\theta}_{s}$ | lg(α) (cm${}^{-1}$) | lg(n) | RMSE${}_{w}$ (cm${}^{3}$cm${}^{-3}$) | |
---|---|---|---|---|---|---|

All | 235 | 0.057 (0.081) | 0.467 (0.139) | −1.606 (0.550) | 0.204 (0.197) | 0.001 (0.002) |

Sands | 112 | 0.050 (0.044) | 0.442 (0.154) | −1.534 (0.421) | 0.321 (0.321) | 0.001 (0.002) |

Loams | 37 | 0.096 (0.137) | 0.525 (0.139) | −1.239 (0.488) | 0.106 (0.109) | 0.001 (0.002) |

Silts | 55 | 0.035 (0.064) | 0.436 (0.097) | −1.890 (0.594) | 0.106 (0.057) | 0.002 (0.003) |

Clays | 31 | 0.071 (0.104) | 0.542 (0.099) | −1.799 (0.644) | 0.075 (0.051) | 0.051 (0.001) |

Measurement | ${\theta}_{r}$ | ${\theta}_{s}$ | lg(α) | lg(n) | RMSE${}_{w}$ |
---|---|---|---|---|---|

field | 0.155 (0.102) | 0.454 (0.196) | −1.562 (0.557 ) | 0.533 (0.354) | 0.015 (0.064) |

lab | 0.075 (0.051) | 0.494 (0.131) | −1.568 (0.311) | 0.288 (0.186) | 0.001 (0.003) |

## 4. Results and Discussion

Code | MEG | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

3000 a | 3000 b | 5000 a | 5000 b | 10,000 a | 10,000 b | 20,000 a | 20,000 b | |||||||||

2763 | 328 | 0.017 | 293 | 0.017 | 399 | 0.017 | 504 | 0.017 | 240 | 0.017 | 310 | 0.017 | 849 | 0.017 | 372 | 0.017 |

2660 | 1117 | 0.013 | 1982 | 0.013 | 4999 | 0.013 | 1578 | 0.013 | 9999 | 0.013 | 9998 | 0.013 | 19,999 | 0.014 | 19,999 | 0.013 |

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Burdine, N.T. Relative permeability calculations from pore-size distribution data. J. Pet. Technol.
**1953**, 5, 71–78. [Google Scholar] [CrossRef] - Mualem, Y. A new model predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef] - Van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef] - Van Genuchten, M.T.; Hopmans, J.W. A Decade of Multidisciplinary Research. Vadose Zone J.
**2013**, 12. [Google Scholar] [CrossRef] - Dettmann, U.; Bechtold, M.; Frahm, E.; Tiemeyer, B. On the applicability of unimodal and bimodal van Genuchten—Mualem based models to peat and other organic soils under evaporation conditions. J. Hydrol.
**2014**, 515, 103–115. [Google Scholar] [CrossRef] - Rawls, W.J.; Pachepsky, Y.; Shen, M.H. Testing soil water retention estimation with the MUUF pedotransfer model using data from the southern United States. J. Hydrol.
**2001**, 251, 177–185. [Google Scholar] [CrossRef] - Pachepsky, Y.A.; van Genuchten, M.T. Pedotransfer functions. Encycl. Agrophysics
**2011**, 556–561. [Google Scholar] - Pan, F.; Pachepsky, Y.; Jacques, D.; Guber, A.; Hill, R.L. Data assimilation with soil water content sensors and pedotransfer functions in soil water flow modeling. Soil Sci. Soc. Am. J.
**2012**, 76, 829–844. [Google Scholar] [CrossRef] - Ramos, T.B.; Gonçalves, M.C.; Brito, D.; Martins, J.C.; Pereira, L.S. Development of class pedotransfer functions for integrating water retention properties into Portuguese soil maps. Soil Res.
**2013**, 51, 262–277. [Google Scholar] [CrossRef] - Schaap, M.G.; Leij, F.J.; van Genuchten, M.T. Neural Network Analysis for Hierarchical Prediction of Soil Hydraulic Properties. Soil Sci. Soc. Am. J.
**1998**, 62, 847–855. [Google Scholar] [CrossRef] - Schaap, M.G.; Feike, J.L. Using neural networks to predict soil water retention and soil hydraulic conductivity. Soil Tillage Res.
**1998**, 47, 37–42. [Google Scholar] [CrossRef] - Schaap, M.G.; Leij, F.J. Improved Prediction of unsaturated hydraulic conductivity with the Mualem-van Genuchten Model. Soil Sci. Soc. Am. J.
**2000**, 64, 843–851. [Google Scholar] [CrossRef] - Schaap, M.G.; Leij, F.J.; van Genuchten, M.T. ROSETTA: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. J. Hydrol.
**2001**, 251, 163–176. [Google Scholar] [CrossRef] - Nemes, A.; Schaap, M.G.; Leij, F.J.; Wösten, J.H.M. Description of the unsaturated soil hydraulic database UNSODA version 2.0. J. Hydrol.
**2001**, 251, 151–162. [Google Scholar] [CrossRef] - Pachepsky, Y.; Pan, F.; Martinez, G. Sensor Network Data Assimilation in Soil Water Flow Modeling. In Application of Soil Physics in Environmental Analyses; Springer International Publishing: Geneva, Switzerland, 2014; pp. 239–260. [Google Scholar]
- Fredlund, M.D.; Wilson, G.W.; Fredlund, D.G. Use of the grain-size distribution for estimation of the soil-water characteristic curve. Can. Geotech. J.
**2002**, 39, 1103–1117. [Google Scholar] [CrossRef] - Haverkamp, R.; Leij, F.J.; Fuentes, C.; Sciortino, A.; Ross, P.J. Soil water retention: I. Introduction of a shape index. Soil Sci. Soc. Am. J.
**2005**, 69, 1881–1890. [Google Scholar] [CrossRef] - Leij, F.J.; Haverkamp, R.; Fuentes, C.; Zatarain, F.; Ross, P.J. Soil water retention: II. Derivation and application of shape index. Soil Sci. Soc. Am. J.
**2005**, 69, 1891–1901. [Google Scholar] [CrossRef] - Bird, N.R.A.; Perrier, E.; Rieu, M. The water retention function for a model of soil structure with pore and solid fractal distributions. Eur. J. Soil Sci.
**2000**, 51, 55–63. [Google Scholar] [CrossRef] - Perfect, E. Modeling the primary drainage curve of prefractal porous media. Vadose Zone J.
**2005**, 4, 959–966. [Google Scholar] [CrossRef] - Huang, G.; Zhang, R. Evaluation of soil water retention curve with the pore-solid fractal model. Geoderma
**2005**, 127, 52–61. [Google Scholar] [CrossRef] - Ghanbarian-Alavijeh, B. Modeling Physical and Hydraulic Properties of Disordered Porous Media: Applications from Percolation Theory and Fractal Geometry. Ph.D. Thesis, Wright State University, Dayton, OH, USA, 2014. [Google Scholar]
- Ghanbarian-Alavijeh, B.; Liaghat, A.; Huang, G.H.; van Genuchten, M.T. Estimation of the van Genuchten soil water retention properties from soil textural data. Pedosphere
**2010**, 20, 456–465. [Google Scholar] [CrossRef] - Lenhard, R.J.; Parker, J.C.; Mishra, S. On the correspondence between Brooks-Corey and van Genuchten models. J. Irrig. Drain. Eng.
**1989**, 115, 744–751. [Google Scholar] [CrossRef] - Kool, J.B.; Parker, J.C.; van Genuchten, M.T. Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: I. Theory and numerical studies. Soil Sci. Soc. Am. J.
**1985**, 49, 1348–1354. [Google Scholar] [CrossRef] - Londra, P.A.; Valiantzas, J.D. Soil water diffusivity determination using a new two-point outflow method. Soil Sci. Soc. Am. J.
**2011**, 75, 1343–1346. [Google Scholar] [CrossRef] - Shao, M.; Horton, R. Integral method for estimating soil hydraulic properties. Soil Sci. Soc. Am. J.
**1998**, 62, 585–592. [Google Scholar] [CrossRef] - Peng, H.; Horton, R.; Lei, T.; Dai, Z.; Wang, X. A modified method for estimating fine and coarse fractal dimensions of soil particle size distributions based on laser diffraction analysis. J. Soils Sediments
**2015**, 15, 937–948. [Google Scholar] [CrossRef] - Kosugi, K. Three-parameter lognormal distribution model for soil water retention. Water Resour. Res.
**1994**, 30, 891–901. [Google Scholar] [CrossRef] - Kosugi, K. General model for unsaturated hydraulic conductivity for soils with lognormal pore-size distribution. Soil Sci. Soc. Am. J.
**1999**, 63, 270–277. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Minimizing the Real Functions of the ICEC’96 Contest by Differential Evolution. In Proceedings of the 1996 IEEE Conference on Evolutionary Computation, Nagoya, Japan, 20–22 May 1996; pp. 842–844.
- César Trejo Zúñiga, E.; López Cruz, I.L.; García, A.R. Parameter estimation for crop growth model using evolutionary and bio-inspired algorithms. Appl. Soft Comput.
**2014**, 23, 474–482. [Google Scholar] [CrossRef] - Storn, R. Differential Evolution Research—Trends and Open Questions. In Advances in Differential Evolution; Chakraborty, U.K., Ed.; Springer-Verlag: Heidelberg, Germany, 2008; pp. 1–32. [Google Scholar]
- Van Genuchten, M.T.; Leij, F.J.; Yates, S.R. The RETC Code for Quantifying the Hydraulic Functions of Unsaturated Soils; EPA Report 600/2-91/065; US Salinity Laboratory, USDA, ARS: Riverside, CA, USA, 1991. [Google Scholar]
- Garg, A.; Vijayaraghavan, V.; Wong, C.H.; Tai, K.; Sumithra, K.; Gao, L.; Singru, P.M. Combined ci-md approach in formulation of engineering moduli of single layer graphene sheet. Simul. Model. Pract. Theory
**2014**, 48, 93–111. [Google Scholar] [CrossRef] - Garg, A.; Tai, K. Stepwise approach for the evolution of generalized genetic programming model in prediction of surface finish of the turning process. Adv. Eng. Softw.
**2014**, 78, 16–27. [Google Scholar] [CrossRef] - Garg, A.; Garg, A.; Tai, K. A multi-gene genetic programming model for estimating stress-dependent soil water retention curves. Comput. Geosci.
**2014**, 18, 45–56. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ou, Z.
Differential Evolution’s Application to Estimation of Soil Water Retention Parameters. *Agronomy* **2015**, *5*, 464-475.
https://doi.org/10.3390/agronomy5030464

**AMA Style**

Ou Z.
Differential Evolution’s Application to Estimation of Soil Water Retention Parameters. *Agronomy*. 2015; 5(3):464-475.
https://doi.org/10.3390/agronomy5030464

**Chicago/Turabian Style**

Ou, Zhonghui.
2015. "Differential Evolution’s Application to Estimation of Soil Water Retention Parameters" *Agronomy* 5, no. 3: 464-475.
https://doi.org/10.3390/agronomy5030464