# Inverse Modeling of Soil Hydraulic Parameters Based on a Hybrid of Vector-Evaluated Genetic Algorithm and Particle Swarm Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{s}, α, n, and K

_{s}) of the van Genuchten–Mualem model has attracted considerable attention. In this study, we proposed a new two-step inversion method, which first estimated the hydraulic parameter θ

_{s}using objective function by the final water content, and subsequently estimated the soil hydraulic parameters α, n, and K

_{s}, using a vector-evaluated genetic algorithm and particle swarm optimization (VEGA-PSO) method based on objective functions by cumulative infiltration and infiltration rate. The parameters were inversely estimated for four types of soils (sand, loam, silt, and clay) under an in silico experiment simulating the tension disc infiltration at three initial water content levels. The results indicated that the method is excellent and robust. Because the objective function had multilocal minima in a tiny range near the true values, inverse estimation of the hydraulic parameters was difficult; however, the estimated soil water retention curves and hydraulic conductivity curves were nearly identical to the true curves. In addition, the proposed method was able to estimate the hydraulic parameters accurately despite substantial measurement errors in initial water content, final water content, and cumulative infiltration, proving that the method was feasible and practical for field application.

## 1. Introduction

_{s}, α, n, and K

_{s}by the objective functions of cumulative infiltration (Q), infiltration rate (v), and final water content (θ

_{final}) using the van Genuchten–Mualem [29] model. Then, based on a hybrid vector-evaluated genetic algorithm (VEGA) [30] and particle swarm optimization (PSO) method [31], we proposed a new inverse method of soil hydraulic parameters named the “two-step method” under in silico experiments of tension disc infiltration, which first searches the hydraulic parameter θ

_{s}by the objective function of θ

_{final}, and then searches the soil hydraulic parameters α, n, and K

_{s}using the hybrid VEGA and PSO method by the objective functions of Q and v. Subsequently, because of the existence of local multiminimum domains of the objective functions in a tiny range of the true values, we analyzed the uniqueness and multiminimum values of the inverse method, as well as the stability and robustness of the inverse method. Finally, we analyzed the feasibility and stability of the proposed method in the presence of measurement errors.

## 2. Theory

#### 2.1. Unsaturated Flow Governing Equation

^{3}cm

^{−3}); h is the matric potential induced by capillary action (cm); K(h) is the hydraulic conductivity (cm min

^{−1}); z is the depth from the soil surface (cm) measured with positive values in a downward direction; r is the radial coordinate (cm); and t is time (min).

#### 2.2. Initial and Boundary Condition

_{initial}) and the initial matric potential (h

_{i}) were the same in the vertical direction. The following initial and boundary equations (Equations (2)–(5)) were used when Equation (1) was solved numerically.

_{i}is the initial matric potential (cm), h

_{0}is the time-variable supply matric potential (cm); and r

_{0}is the disc radius (cm). Equation (1), subject to the abovementioned initial and boundary conditions, was solved using a quasi-three-dimensional finite element model, SWMS-2D, developed by Šimůnek et al. [33]. The numerical solution was based on the mass-conservative iterative scheme proposed by van Genuchten et al. [34].

#### 2.3. van Genuchten–Mualem Model

_{r}and θ

_{s}are the residual and saturated water content levels (cm

^{3}cm

^{−3}), respectively; K

_{s}is the saturated hydraulic conductivity (cm min

^{−1}); α is an empirical parameter that is inversely related to the air-entry pressure value (cm

^{−1}); n is an empirical parameter related to the pore-size distribution; l is an empirical shape parameter, normally equal to 0.5; m = 1 − 1/n; and S

_{e}is the effective saturation.

## 3. Inverse Method

#### 3.1. Formulation of the Inverse Problem

_{e}. In addition, from Equation (8), the parameters of θ

_{r}and θ

_{s}have collinear relationships, thus simultaneous inverse estimation of θ

_{r}and θ

_{s}is not possible. Moreover, the value of θ

_{r}is relatively small with tiny variation range and can be either obtained experimentally (e.g., measuring the water content on very dry soil) or defined as the water content at a large negative value of the matric potential (e.g., the permanent wilting point (h = −15,000 cm)) [29]. Besides, θ

_{r}can also be obtained from the pedotransfer functions by mechanical composition, bulk density, etc. [10]. Therefore, in the following inverse procedure, we set the value of θ

_{r}as constant.

_{s}in the van Genuchten–Mualem model clearly have a complicated nonlinear relationship, and parameters α and K

_{s}vary greatly, which makes them even more difficult to use in an inverse model. Table 1 was obtained from RETC software [34]. Figure 1 illustrates the relationship between α–K

_{s}(1a) and α–n (1b). Notably, as α increases, the corresponding K

_{s}sharply increase, and n increases linearly; the correlation of these parameters should be considered in future research. The ratio α/K

_{s}was used in the development of the inverse model. In Table 1 and other references, the scope of soil hydraulic parameters (θ

_{r}, θ

_{s}, α, n, K

_{s}) and α/K

_{s}were determined in Table 2 (set l as 0.5), which covers the vast majority of soil conditions.

^{*}represents specific measurements, such as Q, v, and θ, at time t

_{i}; Y denotes the corresponding model prediction data under parameter vector β; β is the vector of optimized parameters (α, n, and K

_{s}); and m represents the number of measurement sets (of Q, v, and θ).

_{s}, and K

_{s}, which can be formulated with one of Q, v, or θ in VEGA, or a combination of them in PSO.

#### 3.2. The Hybrid Optimal Algorithm for Inverse Parameters

#### 3.2.1. The Genetic Algorithm and Particle Swarm Optimization

_{i}) in PSO has three characteristics: its position (X

_{i}) represents a potentially optimal solution, its velocity (V

_{i}) represents the direction and speed of the particle’s movement, and its fitness value (F

_{i}) (calculated by the fitness function) represents the relative merit of the particle. The particle moves in the solution space and updates the its location by tracking the current optimal position (P

_{best}) and the current global optimal position (G

_{best}), that is the fitness of the position calculated by one individual and by all particles. In each iteration, each particle is updated; thus, the P

_{best}and G

_{best}collected from the particles in each generation are updated. Each particle’s speed can be adjusted dynamically according to its motion and the motions of the other particles, so that the best solution can be achieved in the optimal solution space.

_{i}) can be updated using the following equations:

_{1}and c

_{2}are learning factors; and r

_{1}and r

_{2}are random numbers uniformly distributed in (0,1).

#### 3.2.2. Multiobjective Optimization Method Using Hybrid VEGA and PSO Algorithms

- Step 1.
- Generate the initial population. The initial population is generated using a binary coding system with total number of K × N; K is the number of subpopulations classified by VEGA, while N is the number of individuals in each subpopulation.
- Step 2.
- Divide the population and calculate fitness in VEGA. VEGA divides the population into K subpopulations and the individuals in each subgroup are evaluated by corresponding fitness functions.
- Step 3.
- Sort and select separately. All the individuals are sorted and selected to eliminate unfit individuals based on the fitness values calculated in Step 2, but different fitness functions operate independently in each subpopulation.
- Step 4.
- Cross and mutate together. The populations selected in Step 3 are mixed together to execute the processes of crossing and mutation, in order to ensure the sufficiency and equilibrium of all fitness functions that VEGA includes.
- Step 5.
- Consider termination of VEGA. If the VEGA iteration number (Steps 3 and 4) is less than the specified value m
_{1}, then return to Step 3; otherwise, go to Step 6 (the PSO process). - Step 6.
- Calculate fitness in PSO. All the individuals are evaluated in PSO by a composite fitness function with several measurement indices, to find minima.
- Step 7.
- Perform PSO operations. Each individual updates its velocity and position according to equations 10 and 11.
- Step 8.
- Consider termination of PSO. If the PSO process iteration number (Steps 6 and 7) is less than the specified value m
_{2}, then return to Step 6; otherwise, return to Step 2 (VEGA process).

_{1}in VEGA and m

_{2}in PSO have been reached. If the process reaches the defined number m

_{3}(Gen), the final termination condition is met and the final solution is output.

#### 3.3. Data Generation

_{initial}, (S

_{e}= 20%, 40% and 60%). The infiltration data used in this study were generated using SWMS-2D software [33]. Only the first three hours of the infiltration process were considered with the time-variable supply matric potential, h

_{0}, which are described in Equation (12). The initial h

_{i}were calculated by the corresponding parameters in Table 1. Figure 3 illustrates the results of Q under θ

_{initial}= 40% effective saturation degree.

#### 3.4. Similarity Evaluation Criteria between Estimated and True SWRC and SWCC

_{i}

^{est}is the ith point of soil water content on the estimated SWRC and SWCC; and θ

_{i}

^{true}is the ith point of soil water content on the true SWRC and SWCC, with the same suction and conductivity.

## 4. Results and Discussion

#### 4.1. Response Surface and Inverse Solutions

_{s}, α, n, and K

_{s}, with the θ

_{r}set as constant. To more clearly analyze the mutual interactions in objective function ψ of the four parameters θ

_{s}, α, n, and K

_{s}, the response surfaces for various parameter planes (α–n, α–K

_{s}, n–K

_{s}, n–θ

_{s}, α–θ

_{s}, and K

_{s}–θ

_{s}) for four types of soil (sand, loam, silt, and clay) with three initial soil water content levels (S

_{e}= 20%, 40%, and 60%) were calculated. Each parameter domain was evenly divided into 50 discrete points, resulting in 2500 (50 × 50) grid points for each response surface. All operations were conducted using an Intel

^{®}XEON

^{®}CPU E5-2683 2.00 GHz processor and 32 GB of RAM in a Windows 7 Ultimate environment; a 28-core server was available for us to compute the algorithm.

#### 4.1.1. Analysis of the Objective Function ψ(θ_{final})

_{final}) was defined using the water content at the end of infiltration (θ

_{final}). Figure 4 illustrates the response surfaces of parameter planes, including α–n (Figure 4a), α–K

_{s}(Figure 4b), n–K

_{s}(Figure 4c), n–θ

_{s}(Figure 4d), α–θ

_{s}(Figure 4e), and K

_{s}–θ

_{s}(Figure 4f), with the remaining two parameters set at the true values in the objective function of ψ(θ

_{final}). For visualization, the logarithmic coordinates were selected for α and K

_{s}. From observing the first three contours in Figure 4a–c, the three planes α–n, α–K

_{s}, n–K

_{s}and the minimum values of ψ(θ

_{final}) all clearly fall in a significant narrow and long valley. This means that even if the other two parameters are determined, the stable optimal values of α, n, and K

_{s}are still difficult to determine from ψ(θ

_{final}). In Figure 4d–f, all three response surfaces, θ

_{s}–n, θ

_{s}–α, and θ

_{s}–K

_{s}, display smooth uniform rings and have obvious extreme points in each minimum circle. This feature suggests that it was possible to use the objective function, ψ(θ

_{final}), to obtain the true value of θ

_{s}.

#### 4.1.2. Analysis of the Objective Functions ψ(v) and ψ(Q)

_{s}can be determined with objective function ψ(θ

_{final}), similar to the abovementioned analysis, the mutual interactions of α–n, α–K

_{s}, and n–K

_{s}can be determined with objective functions ψ(v) and ψ(Q); the results of those calculations are illustrated in Figure 5 and Figure 6. In Figure 5a–c, α–n, α–K

_{s}, and n–K

_{s}are depicted; all response surfaces evidently have global optima; therefore, after the value of θ

_{s}is determined, finding the true values of α, n, and K

_{s}is possible using the optimization method under the objective function ψ(v). Furthermore, the response surface ψ(v), with the parameters α, n, and K

_{s}, was re-estimated near its true value and the results are shown in Figure 5d–f. The response surfaces of α–n (Figure 5d) and n–K

_{s}(Figure 5f) can be observed to have obvious global optimal solutions, whereas for α–K

_{s}(Figure 5e), global optimal solutions exist in the response surfaces with numerous optimal bubbles in the valley. This creates difficulty in precisely searching for the true values of α, n, and K

_{s}only using ψ(v).

_{s}(Figure 6b), and n–K

_{s}(Figure 6c) clearly reveal that the minimum values of ψ(Q) were concentrated in a long narrow valley. Therefore, only using the objective function of ψ(Q) to search for the true values of α, n, and K

_{s}directly is difficult, but it can compress α–n, α–K

_{s}, and n–K

_{s}in a narrow space. Furthermore, similar to the aforementioned analysis, the response surface of ψ(Q) under the parameters α, n, and K

_{s}was recalculated to approximate its true value and the results are shown in Figure 6d–f. This further confirmed that the role of the objective function ψ(Q) is compressing α–n, α–K

_{s}, and n–K

_{s}in a narrow space in the search for α, n, and K

_{s}. Specifically, the response surface of the plane of α–K

_{s}(Figure 6e) indicated that the optimal value was decompressed in a distinct line-like interval, which was helpful for us in searching for the true values.

_{s}(Figure 5f) with the objective function ψ(v), as well as α–K

_{s}(Figure 6e) with the objective function ψ(Q), determining the true values of α, n, and K

_{s}based on the multiple applications of the objective functions ψ(v) and ψ(Q) was possible.

#### 4.2. Inverse Solution and Analysis

#### 4.2.1. The Procedure of Inverse Modeling

_{final}) to search for θ

_{s}with the GA. The values of θ

_{final}for the inverse modeling of typical soils with various θ

_{initial}values were simulated using SWMS-2D. The true parameters are shown in Table 1. For practical use, 24 grid points of θ

_{final}located at radial distance = 0, 5, 10, and 15, and depth = 2, 5, 10, 15, 20, and 35 were used for inverse parameters. According to the scope and features of these parameters, θ

_{s}and n were coded in arithmetic scaling and K

_{s}was coded in logarithmic scaling, whereas α was represented by the ratio value of α/K

_{s}, which was coded in arithmetic scaling. The population size (PopSize) of GA was 24 and the maximum number of iterations was 30.

_{s}set as in the abovementioned inverse model, the hybrid VEGA–PSO algorithm was used with two objective functions, ψ(v) and ψ(Q), comprehensively. The values of Q and v were selected every three minutes, with 60 points in total. The PopSize of PSO was 24, and each subgroup had two objective functions, ψ(v) and ψ(Q). The two subgroups of GA for ψ(v) and ψ(Q) had the same PopSize of 12. The coding mode of θ

_{s}, α, n, and K

_{s}were the same as the aforementioned step. For each individual, we set the generation iterations m

_{1}and m

_{2}as 3 in both GA and PSO for one coupling cycle, and the total number of coupling cycles was set at 30. The selection probability P

_{s}, the crossover probability P

_{c}, and the mutation probability P

_{m}were defined from GA optimization experience as the values 0.667, 0.80, and 0.20, respectively. In PSO, the inertial weight, w, and the learning factors, c

_{1}, and c

_{2}(in Equation (10)), were defined as the default values, equal to 0.729, 2.0, and 2.0 respectively.

#### 4.2.2. Inverse Solution

#### 4.2.3. Analysis of the Inverse Solution

_{s}(Table 3), the true and the estimated SWRCs and SWCCs are almost identical, which indicates that the proposed inverse method is effective and robust.

^{3}·cm

^{−3}, 0.00071–0.00097 cm

^{3}·cm

^{−3}, and 0.0147–0.1603%, respectively) and also for conductivity curves (0.000004–0.00014 cm

^{3}·cm

^{−3}, 0.00003–0.00059 cm

^{3}·cm

^{−3}, and 0.2725–1.2131%, respectively), as well as a large NS (nearly 1.0). Furthermore, the inverse method is indicated to be able to estimate the SWRC and SWCC with precision and robustness.

#### 4.3. Inverse Solutions with Measurement Errors

_{initial}and θ

_{final}. Subsequently, the two random errors were set as 0.02 and 0.05 of standard deviation (2.0% and 5.0%), and superimposed them to the cumulative infiltration. Similar to the error-free inverse method, all of the inversion computations were run again based on the proposed algorithm for the four representative soils (sand, loam, silt, and clay) with the θ

_{initial}(40% degree of effective saturation).

#### 4.3.1. Inverse Solution Analysis Based on Initial and Final Water Content

_{s}, α, n, K

_{s}) of the four typical soils. The results suggested that superimposition of random and system errors on the error-free data result in only small deviations from the true parameters. Table 5 reveals that most values of the inverse parameters were even closer to their true values, which indicates that the proposed method was useful in practical application.

#### 4.3.2. Inverse Solution Analysis Based on Cumulative Infiltration

## 5. Conclusions

_{e}= 20%, 40%, and 60%). A new inverse method called the “two-step method” was proposed. According to the two-step method, the saturated water content, θ

_{s}, was primarily searched through GA with θ

_{final}. Subsequently, using the multiobjective optimization method based on the hybrid VEGA algorithm, the soil characteristic parameters α, n, and K

_{s}could be accurately estimated by cumulative infiltration and infiltration rate obtained by the SWMS-2D software for simulating water flow in two-dimensional variably unsaturated porous media. The results indicated that the proposed method is highly effective. In particular, compared with the traditional multitarget weighted sum optimization method, the hybrid VEGA–PSO algorithm method can overcome the difficulties of weight determination and search for the global optimal values efficiently and robustly.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Bear, J. Dynamic of fluid in porous media. Soil Sci.
**1972**, 120, 174–175. [Google Scholar] - Hillel, D. Fundamentals of Soil Physics; Elsevier Inc.: Amsterdam, The Netherlands, 2013; pp. 387–405. [Google Scholar]
- Durner, W.; Flühler, H. Soil Hydraulic Properties; John Wiley & Sons, Ltd.: New York, NY, USA, 2006; pp. 147–162. [Google Scholar]
- Russo, D. Determining soil hydraulic properties by parameter estimation: On the selection of a model for the hydraulic properties. Water Resour. Res.
**1988**, 24, 453–459. [Google Scholar] [CrossRef] - Klute, A.; Dirksen, C. Hydraulic Conductivity and Diffusivity: Laboratory Methods, 2nd ed.; American Society of Agronomy-Soil Science Society of America: Madison, WI, USA, 1986; pp. 687–734. Available online: https://dl.sciencesocieties.org/publications/books/abstracts/sssabookseries/methodsofsoilan1/687/preview/pdf (accessed on 1 January 1986).
- Schindler, U.; Mueller, L.; Unold, G.V.; Durner, W.; Fank, J. Emerging Measurement Methods for Soil Hydrological Studies; Springer: Berlin, Germany, 2016; pp. 345–363. [Google Scholar]
- Libardi, P.L.; Reichardt, K.; Nielsen, D.R.; Biggar, J.W. Simple field methods for estimating soil hydraulic conductivity. Soil Sci. Soc. Am. J.
**1980**, 44, 3–7. [Google Scholar] [CrossRef] - Ritter, A.; Hupet, F.; Muñoz-Carpena, R.; Lambot, S.; Vanclooster, M. Using inverse methods for estimating soil hydraulic properties from field data as an alternative to direct methods. Agric. Water Manag.
**2003**, 59, 77–96. [Google Scholar] [CrossRef] - Grayson, R.; Bloschl, G. Spatial Patterns in Catchment Hydrology: Observations and Modeling; University Press: Cambridge, UK, 2001; p. 404. [Google Scholar]
- Wösten, J.H.M.; Pachepsky, Y.A.; Rawls, W.J. Pedotransfer functions: Bridging the gap between available basic soil data and missing soil hydraulic characteristics. J. Hydrol.
**2001**, 251, 123–150. [Google Scholar] [CrossRef] - Zachmann, D.W.; Duchateau, P.C.; Klute, A. The Calibration of the Richards Flow Equation for a Draining Column by Parameter Identification1. Soil Sci. Soc. Am. J.
**1981**, 45, 1012–1015. [Google Scholar] [CrossRef] - Zachmann, D.W.; Duchateau, P.C.; Klute, A. Simultaneous Approximation of Water Capacity and Soil Hydraulic Conductivity by Parameter Identification. Soil Sci.
**1982**, 134, 157–163. [Google Scholar] [CrossRef] - Kool, J.B.; Parker, J.C. Analysis of the inverse problem for transient unsaturated flow. Water Resour. Res.
**1988**, 24, 817–830. [Google Scholar] [CrossRef] - Toorman, A.F.; Wierenga, P.J.; Hills, R.G. Parameter estimation of hydraulic properties from one-step outflow data. Water Resour. Res.
**1992**, 28, 3021–3028. [Google Scholar] [CrossRef] - Nor Suhada, A.R.; Askari, M.; Tanaka, T.; Simunek, J.; van Genuchten, M.T. Inverse estimation of soil hydraulic properties under oil palm trees. Geoderma
**2015**, 241–242, 306–312. [Google Scholar] [CrossRef] - Šimůnek, J.; van Genuchten, M.T.; Wendroth, O. Parameter Estimation Analysis of the Evaporation Method for Determining Soil Hydraulic Properties. Soil Sci. Soc. Am. J.
**1998**, 62, 894–905. [Google Scholar] [CrossRef] - Šimůnek, J.; Wendroth, O.; van Genuchten, M.T. Soil Hydraulic Properties from Laboratory Evaporation Experiments by Parameter Estimation. In Proceedings of the International Workshop, Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media, Riverside, CA, USA, 22–24 October 1997; Van Genuchten, M.T., Leij, F.J., Wu, L., Eds.; University of California: Riverside, CA, USA, 1997. [Google Scholar]
- Schindler, U.; Müller, L. Simplifying the evaporation method for quantifying soil hydraulic properties. J. Plant Nutr. Soil Sc.
**2010**, 169, 623–629. [Google Scholar] [CrossRef] - Šimůnek, J.; van Genuchten, M.T. Estimating Unsaturated Soil Hydraulic Properties from Tension Disc Infiltrometer Data by Numerical Inversion. Water Resour. Res.
**1996**, 32, 2683–2696. [Google Scholar] [CrossRef] - Šimůnek, J.; van Genuchten, M.T. Estimating Unsaturated Soil Hydraulic Properties from Multiple Tension Disc Infiltrometer Data. Soil Sci.
**1997**, 162, 383–398. [Google Scholar] [CrossRef] - Gribb, M.M. Parameter Estimation for Determining Hydraulic Properties of a Fine Sand from Transient Flow Measurements. Water Resour. Res.
**1996**, 32, 1965–1974. [Google Scholar] [CrossRef] - Gribb, M.M.; Šimůnek, J.; Leonard, M.F. Development of Cone Penetrometer Method to Determine Soil Hydraulic Properties. J. Geotech. Geoenviron. Eng.
**1998**, 124, 820–829. [Google Scholar] [CrossRef] - Inoue, M.; Šimunek, J.; Hopmans, J.W.; Clausnitzer, V. In situ estimation of soil hydraulic functions using a multistep soil-water extraction technique. Water Resour. Res.
**1998**, 34, 1035–1050. [Google Scholar] [CrossRef] - Dam, J.C.V.; Stricker, J.N.M.; Droogers, P. Inverse Method for Determining Soil Hydraulic Functions from One-Step Outflow Experiments. Soil Sci. Soc. Am. J.
**1992**, 56, 1042–1050. [Google Scholar] - Eching, S.O.; Hopmans, J.W. Optimization of Hydraulic Functions from Transient Outflow and Soil Water Pressure Data. Soil Sci. Soc. Am. J.
**1993**, 57, 1167–1175. [Google Scholar] [CrossRef] - Eching, S.O.; Hopmans, J.W.; Wendroth, O. Unsaturated Hydraulic Conductivity from Transient Multistep Outflow and Soil Water Pressure Data. Soil Sci. Soc. Am. J.
**1994**, 58, 687–695. [Google Scholar] [CrossRef] - Vereecken, H.; Kaiser, R.; Dust, M.; Pütz, T. Evaluation of the Multistep Outflow Method for the Determination of Unsaturated Hydraulic Properties of Soils. Soil Sci.
**1997**, 162, 618–631. [Google Scholar] [CrossRef] - Vrugt, J.A.; Bouten, W.; Weerts, A.H. Information Content of Data for Identifying Soil Hydraulic Parameters from Outflow Experiments. Soil Sci. Soc. Am. J.
**2001**, 65, 19–27. [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] - Konak, A.; Coit, D.W.; Smith, A.E. Multi-objective optimization using genetic algorithms: A tutorial. Reliab. Eng. Syst. Saf.
**2006**, 91, 992–1007. [Google Scholar] [CrossRef] - Eberhart, R.C.; Kennedy, J. A new optimizer using particle swarm theory. In Proceedings of the 6th International Symposium on Micro Machine and Human Science, Nagoya, Japan, 4–6 October 1995; pp. 39–43. [Google Scholar] [CrossRef]
- Richards, L.A. Capillary conduction of liquids through porous mediums. Physics 1931. 1, 318–333. [CrossRef]
- Šimůnek, J.; Vogel, T.; van Genuchten, M.T. The SWMS-2D for Simulating Water Flow and Solute Transport in Two-Dimensional Variably Saturated Media: Version 1.21; US Salinity Laboratory, US Department of Agriculture, Agricultural Research Service: Riverside, CA, USA, 1995.
- Van Genuchten, M.T.; Leij, F.J.; Yates, S.R. The RETC Code for Quantifying the Hydraulic Functions of Unsaturated Soils. Research Report No. EPA/600/2-91/065; US Salinity Laboratory, US Department of Agriculture, Agricultural Research Service: Riverside, CA, USA, 1991. Available online: http://www.pc-progress.com/documents/programs/retc.pdf (accessed on 1 December 1991).
- Holland, J.H. Adaptation in Natural Artificial Systems; The MIT Press: London, UK, 1975. [Google Scholar]
- Schaffer, J.D. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. In Proceedings of the 1st International Conference on Genetic Algorithms, Hillsdale, NJ, USA, 24–26 July 1985; Volume 2, pp. 93–100. [Google Scholar]
- Moriasi, D.N.; Arnold, J.G.; Van Liew, M.W.; Bingner, R.L.; Harmel, R.D.; Veith, T.L. Model Evaluation Guidelines for Systematic Quantification of Accuracy in Watershed Simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef]

**Figure 3.**Cumulative infiltration versus time for four types of soil with three consecutive supply matric potentials (h

_{0}(t = 0) = −15 cm; h

_{0}(t = 60) = −10 cm; and h

_{0}(t = 120) = −3 cm): (

**a**) Sand; (

**b**) Loam; (

**c**) Silt; and (

**d**) Clay.

**Figure 4.**Contours of objective function ψ(θ

_{final}) of silt under θ

_{initial}equal to a 40% effective saturation degree. The hydraulic parameters of silt: θ

_{r}= 0.034 cm

^{3}cm

^{−3}; θ

_{s}= 0.46 cm

^{3}cm

^{−3}; α = 0.016 cm

^{−1}; n = 1.37; K

_{s}= 0.0042 cm min

^{−1}. Results are plotted in the: (

**a**) α–n; (

**b**) α–K

_{s}; (

**c**) n–K

_{s}; (

**d**) n–θ

_{s}; (

**e**) α–θ

_{s}; and (

**f**) K

_{s}–θ

_{s}parameter planes. The solid circles are the true parameters.

**Figure 5.**Contours of objective function ψ(

**v**) of silt under θ

_{initial}equal to a 40% effective saturation degree. The hydraulic parameters of silt: θ

_{r}= 0.034 cm

^{3}cm

^{−3}; θ

_{s}= 0.46 cm

^{3}cm

^{−3}; α = 0.016 cm

^{−1}; n = 1.37; K

_{s}= 0.0042 cm min

^{−1}. Results are plotted in the: (

**a**,

**d**) α–n; (b, e) α–K

_{s}; and (

**c**,

**f**) n–K

_{s}parameter planes. (

**a**–

**c**) depict the global scale, while (

**d**–

**f**) depict the local scale near the minimum. The solid circles are the true parameters.

**Figure 6.**Contours (in detail) of objective function ψ(Q) of silt under θ

_{initial}equal to a 40% effective saturation degree. The hydraulic parameters of silt: θ

_{r}= 0.034 cm

^{3}cm

^{−3}; θ

_{s}= 0.46 cm

^{3}cm

^{−3}; α = 0.016 cm

^{−1}; n = 1.37; K

_{s}= 0.0042 cm min

^{−1}. Results are plotted in the: (

**a**,

**d**) α–n; (

**b**,

**e**) α–K

_{s}; and (

**c**,

**f**) n–K

_{s}parameter planes. (

**a**–

**c**) depict the global scale, while (

**d**–

**f**) depict the local scale near the minimum. The solid circles are the true parameters.

**Figure 7.**Comparison of SWRCs (

**a1**,

**b1**,

**c1**,

**d1**) and conductivity curves (

**a2**,

**b2**,

**c2**,

**d2**) for various treatments (20% effective saturation degree for initial water content, circle; 40% effective saturation degree for initial water content, triangle; 60% effective saturation degree for initial water content, square; true value, line): (

**a**) Sand; (

**b**) Loam; (

**c**) Silt; and (

**d**) Clay.

**Figure 8.**Comparison of SWRCs (

**a1**,

**b1**,

**c1**,

**d1**) and conductivity curves (

**a2**,

**b2**,

**c2**,

**d2**) with random errors (RE) and system errors (SE) for various treatments with objective function ψ(θ

_{initial}): (

**a**) Sand; (

**b**) Loam; (

**c**) Silt; and (

**d**) Clay.

**Figure 9.**Comparison of SWRCs (

**a1**,

**b1**,

**c1**,

**d1**) and conductivity curves (

**a2**,

**b2**,

**c2**,

**d2**) with random errors (RE) and system errors (SE) for various treatments with objective function ψ(θ

_{final}): (

**a**) Sand; (

**b**) Loam; (

**c**) Silt; and (

**d**) Clay.

**Figure 10.**Comparison of SWRCs (

**a1**,

**b1**,

**c1**,

**d1**) and conductivity curves (

**a2**,

**b2**,

**c2**,

**d2**) with random errors and system errors for various treatments with objective function ψ(Q) (random error (RE), 2.0%, circle; random error, 5.0%, triangle; true value, line): (

**a**) Sand; (

**b**) Loam; (

**c**) Silt; and (

**d**) Clay.

Texture Class | θ_{r} | θ_{s} | α | n | l | K_{s} | α/K_{s} |
---|---|---|---|---|---|---|---|

cm^{3} cm^{−3} | cm^{3} cm^{−3} | cm^{−1} | - | - | cm min^{−1} | ||

Sand | 0.045 | 0.43 | 0.145 | 2.68 | 0.5 | 0.49500 | 0.293 |

Loamy sand | 0.057 | 0.41 | 0.124 | 2.28 | 0.5 | 0.24317 | 0.510 |

Sandy loam | 0.065 | 0.41 | 0.075 | 1.89 | 0.5 | 0.07367 | 1.018 |

Loam | 0.078 | 0.43 | 0.036 | 1.56 | 0.5 | 0.01733 | 2.077 |

Silt | 0.034 | 0.46 | 0.016 | 1.37 | 0.5 | 0.00417 | 3.840 |

Silt loam | 0.067 | 0.45 | 0.020 | 1.41 | 0.5 | 0.00750 | 2.667 |

Sandy clay loam | 0.100 | 0.39 | 0.059 | 1.48 | 0.5 | 0.02183 | 2.702 |

Clay loam | 0.095 | 0.41 | 0.019 | 1.31 | 0.5 | 0.00433 | 4.385 |

Silty clay loam | 0.089 | 0.43 | 0.010 | 1.23 | 0.5 | 0.00117 | 8.571 |

Sandy clay | 0.100 | 0.38 | 0.027 | 1.23 | 0.5 | 0.00200 | 13.500 |

Silty clay | 0.070 | 0.36 | 0.005 | 1.09 | 0.5 | 0.00033 | 15.000 |

Clay | 0.068 | 0.38 | 0.008 | 1.09 | 0.5 | 0.00333 | 2.400 |

_{r}, θ

_{s}, α, n, l, and K

_{s}were obtained from RETC.

Scope | θ_{s} | α | n | K_{s} | α/K_{s} |
---|---|---|---|---|---|

- | cm^{−1} | - | cm min^{−1} | - | |

min | 0.35 | 0.001 | 1.05 | 0.00010 | 0.2 |

max | 0.55 | 0.200 | 3.00 | 0.60000 | 20 |

Soil | Category | θ_{r} | θ_{s} | α | n | K_{s} |
---|---|---|---|---|---|---|

cm^{3} cm^{−3} | cm^{3} cm^{−3} | cm^{−1} | - | cm min^{−1} | ||

Sand | 20% | 0.0450 | 0.4303 | 0.1455 | 2.6967 | 0.4922 |

40% | 0.4314 | 0.1449 | 2.6751 | 0.4971 | ||

60% | 0.4285 | 0.1448 | 2.6967 | 0.4898 | ||

True Value | 0.0450 | 0.4300 | 0.1450 | 2.6800 | 0.4950 | |

Loam | 20% | 0.0780 | 0.4302 | 0.0359 | 1.5196 | 0.0181 |

40% | 0.4285 | 0.0356 | 1.5522 | 0.0170 | ||

60% | 0.4299 | 0.0363 | 1.5798 | 0.0171 | ||

True Value | 0.0780 | 0.4300 | 0.0360 | 1.5600 | 0.0173 | |

Silt | 20% | 0.0340 | 0.4606 | 0.0161 | 1.3739 | 0.0042 |

40% | 0.4623 | 0.0163 | 1.3639 | 0.0044 | ||

60% | 0.4594 | 0.0166 | 1.3905 | 0.0041 | ||

True Value | 0.0340 | 0.4600 | 0.0160 | 1.3700 | 0.0042 | |

Clay | 20% | 0.0680 | 0.3787 | 0.0088 | 1.0889 | 0.0039 |

40% | 0.3812 | 0.0069 | 1.0919 | 0.0027 | ||

60% | 0.3803 | 0.0084 | 1.0919 | 0.0033 | ||

True Value | 0.0680 | 0.3800 | 0.0080 | 1.0900 | 0.0033 |

Soil | From h(θ) | From K(θ) | ||||||
---|---|---|---|---|---|---|---|---|

MAE | RMSE | PBIAS (%) | NS | MAE | RMSE | PBIAS (%) | NS | |

Sand | 0.00069 | 0.00082 | −0.0147 | 0.9999 | 0.00014 | 0.00059 | 0.4023 | 0.9999 |

Loam | 0.00039 | 0.00060 | 0.0987 | 0.9999 | 0.00002 | 0.00008 | −0.5136 | 0.9991 |

Silt | 0.00076 | 0.00097 | −0.1603 | 0.9999 | 0.00000 | 0.00003 | −1.2131 | 0.9985 |

Clay | 0.00060 | 0.00071 | −0.0161 | 0.9999 | 0.00002 | 0.00008 | −0.2725 | 0.9778 |

**Table 5.**Estimated parameters considering random and system errors on initial and final water content levels for the four typical soils.

Texture Class | Error Source | Error Category | θ_{r} | θ_{s} | α | n | K_{s} | |
---|---|---|---|---|---|---|---|---|

cm^{3} cm^{−3} | cm^{3} cm^{−3} | cm^{−1} | - | cm min^{−1} | ||||

Sand | θ_{initial} | RE | 5.0% | 0.0450 | 0.4292 | 0.1430 | 2.6512 | 0.4912 |

SE | θ_{i} + 0.02 | 0.4352 | 0.1389 | 2.6931 | 0.4959 | |||

θ_{i} − 0.02 | 0.4096 | 0.1281 | 2.6988 | 0.4912 | ||||

θ_{final} | RE | 5.0% | 0.4296 | 0.1455 | 2.6780 | 0.4867 | ||

SE | θ_{f} + 0.02 | 0.4423 | 0.1464 | 2.6874 | 0.4943 | |||

θ_{f} − 0.02 | 0.4004 | 0.1449 | 2.6994 | 0.4877 | ||||

True Parameter | 0.0450 | 0.4300 | 0.1450 | 2.6800 | 0.4950 | |||

Loam | θ_{initial} | RE | 5.0% | 0.0780 | 0.4435 | 0.0376 | 1.5437 | 0.0185 |

SE | θ_{i} + 0.02 | 0.4712 | 0.0404 | 1.5799 | 0.0189 | |||

θ_{i} − 0.02 | 0.3866 | 0.0341 | 1.5151 | 0.0191 | ||||

θ_{final} | RE | 5.0% | 0.4266 | 0.0365 | 1.5265 | 0.0188 | ||

SE | θ_{f} + 0.02 | 0.4798 | 0.0362 | 1.5761 | 0.0164 | |||

θ_{f} − 0.02 | 0.3893 | 0.0328 | 1.5151 | 0.0162 | ||||

True Parameter | 0.0780 | 0.4300 | 0.0360 | 1.5600 | 0.0173 | |||

Silt | θ_{initial} | RE | 5.0% | 0.0340 | 0.4546 | 0.0159 | 1.3768 | 0.0040 |

SE | θ_{i} + 0.02 | 0.5035 | 0.0233 | 1.4474 | 0.0067 | |||

θ_{i} − 0.02 | 0.4218 | 0.0136 | 1.2787 | 0.0028 | ||||

θ_{final} | RE | 5.0% | 0.4640 | 0.0167 | 1.3493 | 0.0046 | ||

SE | θ_{f} + 0.02 | 0.5072 | 0.0179 | 1.2825 | 0.0056 | |||

θ_{f} − 0.02 | 0.4051 | 0.0143 | 1.2177 | 0.0033 | ||||

True Parameter | 0.0340 | 0.4600 | 0.0160 | 1.3700 | 0.0042 | |||

Clay | θ_{initial} | RE | 5.0% | 0.0680 | 0.3965 | 0.0087 | 1.0854 | 0.0036 |

SE | θ_{i} + 0.02 | 0.4241 | 0.0105 | 1.1220 | 0.0051 | |||

θ_{i} − 0.02 | 0.3522 | 0.0062 | 1.0729 | 0.0022 | ||||

θ_{final} | RE | 5.0% | 0.3774 | 0.0079 | 1.0824 | 0.0032 | ||

SE | θ_{f} + 0.02 | 0.4296 | 0.0112 | 1.1015 | 0.0056 | |||

θ_{f} − 0.02 | 0.3514 | 0.0054 | 1.0681 | 0.0018 | ||||

True Parameter | 0.0680 | 0.3800 | 0.0080 | 1.0900 | 0.0033 |

**Table 6.**Soil water content errors from estimated SWRCs and SWRCs considering both random and system errors on initial and final water content levels for the four typical soils.

Source | Texture Class | Error Category | From h(θ) | From K(θ) | |||||
---|---|---|---|---|---|---|---|---|---|

RMSE | PBIAS | NS | RMSE | PBIAS | NS | ||||

cm^{3} cm^{−3} | % | - | cm^{3} cm^{−3} | % | - | ||||

θ_{initial} | Sand | RE | 5.0% | 0.0018 | 0.5904 | 0.9999 | 0.0004 | −0.1621 | 1.0000 |

SE | θ_{i} + 0.02 | 0.0057 | 2.3078 | 0.9986 | 0.0025 | 1.0377 | 0.9995 | ||

θ_{i} − 0.02 | 0.0090 | −0.1401 | 0.9963 | 0.0096 | −4.0328 | 0.9924 | |||

Loam | RE | 5.0% | 0.0065 | 2.4456 | 0.9976 | 0.0064 | 2.4012 | 0.9961 | |

SE | θ_{i} + 0.02 | 0.0176 | 4.4951 | 0.9826 | 0.0195 | 7.3571 | 0.9630 | ||

θ_{i} − 0.02 | 0.0200 | −5.0526 | 0.9777 | 0.0205 | −7.7544 | 0.9589 | |||

Silt | RE | 5.0% | 0.0035 | −1.5893 | 0.9995 | 0.0026 | −1.0461 | 0.9996 | |

SE | θ_{i} + 0.02 | 0.0243 | −3.6490 | 0.9771 | 0.0205 | 8.4125 | 0.9719 | ||

θ_{i} − 0.02 | 0.0254 | 4.5768 | 0.9752 | 0.0181 | −7.4018 | 0.9783 | |||

Clay | RE | 5.0% | 0.0098 | 5.0416 | 0.9905 | 0.0078 | 3.3498 | 0.9924 | |

SE | θ_{i} + 0.02 | 0.0209 | −5.1589 | 0.9565 | 0.0208 | 8.9355 | 0.9461 | ||

θ_{i} − 0.02 | 0.0142 | 3.5927 | 0.9799 | 0.0132 | −5.6430 | 0.9785 | |||

θ_{final} | Sand | RE | 5.0% | 0.0004 | −0.1770 | 1.0000 | 0.0002 | −0.0847 | 1.0000 |

SE | θ_{i} + 0.02 | 0.0056 | 1.9248 | 0.9986 | 0.0058 | 2.4310 | 0.9973 | ||

θ_{i} − 0.02 | 0.0149 | −5.7688 | 0.9901 | 0.0140 | −5.8520 | 0.9841 | |||

Loam | RE | 5.0% | 0.0034 | 0.7115 | 0.9994 | 0.0016 | −0.6000 | 0.9998 | |

SE | θ_{i} + 0.02 | 0.0244 | 7.7015 | 0.9666 | 0.0235 | 8.8927 | 0.9460 | ||

θ_{i} − 0.02 | 0.0182 | −4.1199 | 0.9815 | 0.0192 | −7.2658 | 0.9639 | |||

Silt | RE | 5.0% | 0.0047 | 2.2844 | 0.9991 | 0.0019 | 0.7703 | 0.9998 | |

SE | θ_{i} + 0.02 | 0.0385 | 19.0410 | 0.9428 | 0.0223 | 9.1314 | 0.9669 | ||

θ_{i} − 0.02 | 0.0433 | 10.3832 | 0.9275 | 0.0259 | −10.6175 | 0.9553 | |||

Clay | RE | 5.0% | 0.0063 | 3.0968 | 0.9961 | 0.0012 | −0.5325 | 0.9998 | |

SE | θ_{i} + 0.02 | 0.0169 | 2.7096 | 0.9715 | 0.0234 | 10.0447 | 0.9319 | ||

θ_{i} − 0.02 | 0.0188 | 6.7122 | 0.9646 | 0.0135 | −5.8014 | 0.9773 |

**Table 7.**Estimated parameters considering random errors on cumulative infiltration for the four typical soils.

Texture Class | Error Category | θ_{r} | θ_{s} | α | n | K_{s} |
---|---|---|---|---|---|---|

cm^{3} cm^{−3} | cm^{3} cm^{−3} | cm^{−1} | - | cm min^{−1} | ||

Sand | 2.0% | 0.0450 | 0.4314 | 0.1416 | 2.6903 | 0.4930 |

5.0% | 0.4314 | 0.1475 | 2.6988 | 0.4987 | ||

True Parameter | 0.0450 | 0.4300 | 0.1450 | 2.6800 | 0.4950 | |

Loam | 2.0% | 0.0780 | 0.4285 | 0.0341 | 1.6466 | 0.0144 |

5.0% | 0.4285 | 0.0365 | 1.7919 | 0.0124 | ||

True Parameter | 0.0780 | 0.4300 | 0.0360 | 1.5600 | 0.0173 | |

Silt | 2.0% | 0.0340 | 0.4623 | 0.0147 | 1.2996 | 0.0043 |

5.0% | 0.4623 | 0.0160 | 1.3817 | 0.0039 | ||

True Parameter | 0.0340 | 0.4600 | 0.0160 | 1.3700 | 0.0042 | |

Clay | 2.0% | 0.0680 | 0.3812 | 0.0104 | 1.0862 | 0.0046 |

5.0% | 0.3812 | 0.0074 | 1.0901 | 0.0027 | ||

True Parameter | 0.0680 | 0.3800 | 0.0080 | 1.0900 | 0.0033 |

**Table 8.**Soil water content errors from estimated SWRCs and SWCCs considering random errors on cumulative infiltration for the four typical soils.

Texture Class | Error Category | From h(θ) | From K(θ) | ||||
---|---|---|---|---|---|---|---|

RMSE | PBIAS | NS | RMSE | PBIAS | NS | ||

cm^{3}cm^{−3} | % | - | cm^{3}cm^{−3} | % | - | ||

Sand | 2.0% | 0.0024 | 0.9524 | 0.9997 | 0.0007 | 0.2774 | 0.9997 |

5.0% | 0.0017 | −0.4835 | 0.9999 | 0.0007 | 0.2774 | 0.9999 | |

Loam | 2.0% | 0.0073 | −2.6803 | 0.9970 | 0.0007 | −0.2678 | 1.0000 |

5.0% | 0.0205 | −7.2361 | 0.9765 | 0.0007 | −0.2678 | 1.0000 | |

Silt | 2.0% | 0.0212 | 9.5425 | 0.9826 | 0.0011 | 0.4451 | 0.9999 |

5.0% | 0.0023 | −0.6324 | 0.9998 | 0.0011 | 0.4451 | 0.9999 | |

Clay | 2.0% | 0.0022 | 0.9520 | 0.9995 | 0.0006 | 0.2432 | 1.0000 |

5.0% | 0.0011 | 0.4907 | 0.9999 | 0.0006 | 0.2432 | 1.0000 |

© 2018 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.-B.; Liu, Y.; Nie, W.-B.; Ma, X.-Y.
Inverse Modeling of Soil Hydraulic Parameters Based on a Hybrid of Vector-Evaluated Genetic Algorithm and Particle Swarm Optimization. *Water* **2018**, *10*, 84.
https://doi.org/10.3390/w10010084

**AMA Style**

Li Y-B, Liu Y, Nie W-B, Ma X-Y.
Inverse Modeling of Soil Hydraulic Parameters Based on a Hybrid of Vector-Evaluated Genetic Algorithm and Particle Swarm Optimization. *Water*. 2018; 10(1):84.
https://doi.org/10.3390/w10010084

**Chicago/Turabian Style**

Li, Yi-Bo, Ye Liu, Wei-Bo Nie, and Xiao-Yi Ma.
2018. "Inverse Modeling of Soil Hydraulic Parameters Based on a Hybrid of Vector-Evaluated Genetic Algorithm and Particle Swarm Optimization" *Water* 10, no. 1: 84.
https://doi.org/10.3390/w10010084