# Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: II. Evaluation of Parametric Pedotransfer Functions Against Direct Fits

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{m}model). In addition, eleven bimodal, Peters–Durner–Iden (PDI) and bimodal-PDI variants of the original expressions were studied. Six modeling scenarios (S1 to S6) were examined with different combinations of the following input predictors: soil texture (percentages of sand, silt and clay), soil bulk density, organic matter content, percent of stable aggregates and saturated water content (θ

_{s}). Although a majority of the model parameters showed low correlations with basic soil properties, most of the parametric PTFs provided reasonable water content estimations. The VG

_{m}parametric PTF with an RMSE of 0.034 cm

^{3}cm

^{−3}was the best PTF when all input predictors were considered. When averaged across modeling scenarios, the PDI variant of the K model with an RMSE of 0.045 cm

^{3}cm

^{−3}showed the highest performance. The best performance of all models occurred at S6 when θs was considered as an additional input predictor. The second-best performance for 11 out of the 16 models belonged to S1 with soil textural components as the only inputs. Our results do not recommend the development of parametric PTFs using bimodal variants because of their poor performance, which is attributed to their high number of free parameters.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Collection and Laboratory Analyses

#### 2.2. SWRC Parametrization

_{m}model; [24]).

_{m}models. HYPROP-FIT software was used to fit the soil hydraulic models to the measured water retention data. All models are explained in detail, and equations are provided in Appendix A.

#### 2.3. Developing Parametric Pedotransfer Functions

_{s}) and SA were considered as additional inputs.

#### 2.4. Evaluation Criteria

^{3}cm

^{−3}), respectively; $\overline{M}$ and $\overline{E}$ are the mean measured and the mean estimated water content values, and n is the total number of measured water retention points for all 81 samples (n = 7729). The evaluation statistics were calculated only for the test subset after combining the PTF estimations for the 8 folds resulting in a total of 96 values (16 models × 6 scenarios) for each statistic.

## 3. Results

#### 3.1. Overall Performance of the Models

^{3}cm

^{−3}to 0.100 cm

^{3}cm

^{−3}. The MBE values range from −0.012 to 0.059 with bimodal variants (except FX-b-PDI) showing roughly an order of magnitude greater MBE values than unimodal expressions. The lowest and highest MAE values of 0.024 cm

^{3}cm

^{−3}and 0.079 cm

^{3}cm

^{−3}belong to the VG

_{m}model (S6) and the VG

_{m}bimodal model (S4), respectively. When averaged across the scenarios, the K PDI and the VG

_{m}bimodal models show the lowest and the highest MAE values of 0.036 cm

^{3}cm

^{−3}and 0.072 cm

^{3}cm

^{−3}, respectively. When the MAE is calculated for each sample separately, the VG

_{m}bimodal provides the worst fit for 24% to 61% of the samples across scenarios.

^{3}cm

^{−3}and 0.037 cm

^{3}cm

^{−3}, respectively. The minimum MAE values for all models belong to S6. The second-best performance for 11 out of the 16 models belongs to S1. The maximum MAE values belong to multiple scenarios for different models with S2 and S5 having the highest MAE values for a total of ten and four models, respectively.

_{m}(MAE = 0.024 cm

^{3}cm

^{−3}), the VG (MAE = 0.026 cm

^{3}cm

^{−3}), the VG PDI (MAE = 0.026 cm

^{3}cm

^{−3}), the BC (MAE = 0.027 cm

^{3}cm

^{−3}) and the K PDI (MAE = 0.027 cm

^{3}cm

^{−3}). The first to fifth worst models are the VG

_{m}bimodal (MAE = 0.064 cm

^{3}cm

^{−3}), the VG bimodal (MAE = 0.064 cm

^{3}cm

^{−3}), the VG bimodal-PDI (MAE = 0.046 cm

^{3}cm

^{−3}), the K bimodal (MAE = 0.045 cm

^{3}cm

^{−3}) and the VG

_{m}bimodal-PDI (MAE = 0.044 cm

^{3}cm

^{−3}). When the MAE is calculated for each sample separately, the VG

_{m}bimodal followed by the VG bimodal and the VG bimodal-PDI provide the worst fit for 56%, 13%, and 13% of the samples, respectively. The VG PDI, the K PDI, and the VG provide the best fit for 16%, 13% and 11% of the samples, respectively.

_{s}and θ

_{r}.

#### 3.2. Performance across Textural and Tension Classes

^{3}cm

^{−3}to 0.06 cm

^{3}cm

^{−3}. A similar trend is observed for the PDI variants. However, the bimodal and bimodal-PDI variants, in particular variants of the K, the VG and the VG

_{m}models, show relatively larger MAE values as well as greater degrees of fluctuations across tensions classes. For the VG

_{m}bimodal, the K bimodal and the VG

_{m}bimodal-PDI models, MAE values peak at the wet and the dry parts of the PTF. For the VG PDI and the VG bimodal-PDI, MAE values increase from the wet-end up to the intermediate tension range but level off toward the dry part.

## 4. Discussion

#### 4.1. Overall Performance of the Parametric PTFs

^{3}cm

^{−3}to 0.085 cm

^{3}cm

^{−3}[11]. Therefore, PTFs developed using the HYPROP data, both in this study and by Haghverdi et al. [18], rank high among previously published parametric PTFs. Our results (Table 6), however, show that a majority of the SWRC model parameters cannot be estimated with high accuracy as a function of basic soil properties. The low correlations and differences between parametric PTFs estimated versus fitted soil hydraulic parameters were also reported by other researchers [20,38]. Tomasella et al. [21] argued that the relationships between the basic soil properties and the model parameters are often complicated and also vary at different parts of the curve, which makes them difficult to estimate using parametric PTFs. Schaap et al. [20] mentioned that macroscopic variables typically used as inputs for parametric PTFs could not predict information contained by the SWRC.

_{m}parametric PTF with an RMSE of 0.034 cm

^{3}cm

^{−3}was the best PTF when all input predictors were considered. When averaged across modeling scenarios with different combinations of input predictors, the K-PDI model with an RMSE of 0.045 cm

^{3}cm

^{−3}showed the highest performance. This performance is also comparable to the performance of neural network PC-PTFs developed using similar soils with RMSE values of 0.033 cm

^{3}cm

^{−3}and 0.045 cm

^{3}cm

^{−3}with the HYPROP and sandbox/pressure plates data, respectively [5,18].

#### 4.2. Performance Across Soil Textures and Tension Classes

#### 4.3. Importance of the Input Parameters

^{3}cm

^{−3}) and SOM (S3: average RMSE: 0.068 cm

^{3}cm

^{−3}) were considered as input predictors in addition to soil texture (S1: average RMSE: 0.061 cm

^{3}cm

^{−3}). When averaged across the models, the first five scenarios (S1–S5) showed a relatively similar performance with average RMSE values varying between 0.061 to 0.068 cm

^{3}cm

^{−3}. The best performance of all models occurred at S6 (average RMSE: 0.052 cm

^{3}cm

^{−3}) when θ

_{s}was added as an additional input predictor. The low effect of SOM and the high impact of θ

_{s}as input predictors were also reported for PC-PTFs developed using similar soils by Haghverdi et al. [18]. The low impact of SOM is attributed to the fact that a majority of the soils used in the two studies were collected from the dry Central Anatolia Region of Turkey with low organic matter content. Haghverdi et al. [4] also reported only a slight improvement in the performance of their PTFs when OC and BD were added as extra predictors. Cornelis et al. [2] studied multiple parametric PTFs and observed no correlation between SOM, BD and the performance of the PTFs. Vereecken et al. [11] reviewed several studies on van Genuchten based parametric PTFs and noticed improvements in the performance of parametric PTFs when the water content data were considered as input in addition to the basic soil properties.

_{s}as an input predictor remarkably decreased the accuracy of the most PTFs down to unacceptable performance levels (RMSE values ranged from 0.061 to 0.159 cm

^{3}cm

^{−3}). PTFs without θ

_{s}as input performed much better in this study (average RMSE values ranged from 0.061 to 0.068 cm

^{3}cm

^{−3}) compared to Haghverdi et al. [18], which is a promising result given the practical difficulties to measure θs directly. We attribute this to the differences between the structure of the PC-PTFs developed by Haghverdi et al. [18] and the parametric PTFs derived in this study. The PC-PTF relies on neural networks to relate basic soil properties to the whole SWRC, whereas in parametric PTFs, soil hydraulic models govern the shape and continuity of the estimated SWRC. Consequently, including θ

_{s}as a measure of the soil water content upper boundary seems to be much more critical when PC-PTFs are developed using the HYPROP data.

#### 4.4. Fitted versus Parametric PTF Estimated SWRC

^{3}cm

^{−3}) versus PTF-estimated (S6, average MAE: 0.037 cm

^{3}cm

^{−3}) parameters. Although bimodal and bimodal-PDI expressions provided the best fit (average MAE: 0.003 cm

^{3}cm

^{−3}) to the measured data, their parametric PTFs ranked lowest (S6, average MAE: 0.048 cm

^{3}cm

^{−3}); hence, application of their PTFs is not recommended. The bimodal-PDI and bimodal variants provided a similar direct fit performance as the PDI variants did, but their parametric PTFs showed a much lesser accuracy (S6, average MAE: 0.048 cm

^{3}cm

^{−3}) compared to PDI models (S6, average MAE: 0.029 cm

^{3}cm

^{−3}).

^{3}cm

^{−3}), but their parametric PTFs on average performed as high as the PDI variants (S6, average MAE: 0.029 cm

^{3}cm

^{−3}). In the first part of this study [25], we showed that the PDI variants are suggested for full SWRC parametrization using HYPROP data due to their superior dry-end performance (average MAE: 0.017 cm

^{3}cm

^{−3}) compared to the traditional unimodal expressions (average MAE: 0.041 cm

^{3}cm

^{−3}). Therefore, the combined results of both parts of this study suggest the application of PDI variants for full SWRC parametrization (direct fit and parametric PTFs) using HYPROP data. More investigation is needed to determine whether these findings hold for different data sets, parametric PTFs developed using data mining techniques and the nonparametric k-nearest PTF of Haghverdi et al. [3].

## 5. Conclusions

^{3}cm

^{−3}). The parametric PTF of the PDI variant [30,31] of the Kosugi model [36] showed the highest average performance across six modeling scenarios with different combinations of input predictors (0.045 cm

^{3}cm

^{−3}). The high number of model parameters of bimodal variants negatively impacted the performance of their parametric PTFs. PDI variants, on the other hand, have the same number of free parameters as original unimodal and showed an excellent fit to the complete SWRC [25], and their parametric PTFs highly ranked among all models. Consequently, combined results of both parts of this study recommend the application of PDI variants and FX model for estimation and parametrization of the complete SWRC (i.e., from saturation to oven dryness) using typical data obtained by the HYPROP system.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Unimodal Expressions:

_{r}and θ

_{s}are the residual and saturated soil water contents, respectively.

_{r}is the tension corresponding to θ

_{r}, h

_{0}is the soil tension at zero water content, and e is the Euler number.

_{m}is the suction corresponding to the median pore radius.

_{m}model [24] with m as an additional shape parameter has five free parameters:

#### Appendix A.2. PDI Expressions:

^{cap}(h), and an adsorptive retention term, θ

^{ad}(h), and is given as:

^{cap}and S

^{ad}are the capillary and the water adsorption saturation functions, and θ

_{r}is the maximum water content for the water adsorption.

_{0}, the S

^{cap}is substituted by scaled versions of the original functions:

_{0}.

_{a}and x

_{0}are pF values at suctions equal to h

_{a}and h

_{0}, respectively, h

_{a}is the suction at air entry for the adsorptive retention, and b is the shape parameter, which is given by:

#### Appendix A.3. Bimodal Expressions:

^{ad}= 1):

_{i}is the weighting factor for the subfunction i, subject to 0 < w

_{i}< 1 and Σ w

_{i}= 1. The $\Gamma \left(h\right)$ is calculated using Equations (10)–(13).

#### Appendix A.4. Bimodal-PDI Expressions:

^{ad}is calculated using Equation (14) for which the shape parameter b is calculated using Equations (15) and (16). The shape parameter is calculated only for the “coarsest” subfunction, which is the subfunction with the lowest h

_{m}value for the K model or the highest α value for the FX, the VG and the VG

_{m}models.

## Appendix B

**Table A1.**List of regression equations (parametric PTFs) for estimating the parameters of the selected soil water retention models for the six modeling scenarios, S1–S6 (bold text).

par | Equation | |
---|---|---|

BC | log(α) | S1, S2, S4→ −0.959 − 4.542(Silt) − 2.868(Clay^{2}) + 11.817(Clay)(Silt)S3→ −1.009 − 4.614(Silt) + 0.050(SOM) − 2.838(Clay^{2}) + 11.885(Clay)(Silt)S5→ −0.663 − 1.940(Silt) − 0.321(BD^{2}) + 1.402(Clay^{2}) + 0.063(BD)(SOM) − 0.133(Clay)(SOM)S6→ −1.273 − 3.199(Silt^{2}) − 2.670(Clay^{2}) − 1.419(BD)(Silt) + 9.453(Clay)(Silt) |

λ | S1, S4→ 0.398 + 0.003(Silt) + 0.682(Clay)(Silt)S2→ 0.506 + 0.477(Clay) − 0.777(BD)(Clay) − 0.086(BD)(Silt)S3→ 0.371 − 0.113(Clay)(SOM) + 0.004(Silt)(SOM)S5→ 0.501 + 0.561(Clay) + 0.001(SA) − 1.84 × 10 ^{− 5}(SA^{2}) − 0.833(BD)(Clay) − 0.086 (Silt)(SOM)S6→ 0.983 − 0.754(BD)(θ_{S}) − 0.284(BD)(Clay) | |

θr | S1, S3, & S4→ 0.028 − 0.194(Silt^{2}) + 0.434(Clay^{2})S2 & S5→ 0.308 − 0.109(BD^{2}) − 0.055(Clay^{2}) − 0.165(BD)(Silt)S6→ 0.068 + 0.640(θ_{S}^{2}) − 0.283(BD)(θ_{S}) − 0.066(Silt)(SOM) | |

θs | S1 & S4→ 0.446 + 1.358(Clay)(Silt)S2→ 0.781 − 0.135(BD^{2}) + 0.124(Clay^{2})S3→ 0.305 + 1.319(Silt) − 1.895(Silt^{2}) + 0.178(Clay)(SOM)S5→ 0.694 + 0.085(SOM) − 0.157(BD^{2}) + 7.06 × 10^{−6}(SA^{2}) + 0.264(BD)(Clay) − 0.187(Clay)(SOM) − 0.001(Silt)(SA) | |

FX | Log(α) | S1, S4→ −2.036 − 6.063(Silt^{2}) − 1.077(Clay^{2}) + 9.015(Clay)(Silt)S2→ −1.856 + 1.671(BD)(Clay) − 1.764(BD)(Silt)S3→ −1.894 − 4.431(Silt^{2}) − 1.573(Clay^{2}) + 9.542(Clay)(Silt) − 0.773(Silt)(SOM)S5→ −1.769 − 5.589(Silt^{2}) − 1.268(Clay^{2}) − 0.149(BD)(SOM) + 8.405(Clay)(Silt)S6→ −1.555 − 8.899(Silt^{2}) − 0.121(BD)(SOM) − 5.501(θ_{S})(Clay) + 18.627(Clay)(Silt) |

Log(n) | S1, S3→ 0.319 − 0.758(Silt^{2}) + 1.533(Clay^{2})S2→ 1.051 + 4.176(Silt^{2}) − 2.709(BD)(Silt)S4→ 0.475 − 0.541(Silt) + 0.012(Clay)(SA)S5, S6→ 0.767 + 0.454(SOM) − 0.587(BD)(Silt) − 0.426(BD)(SOM) | |

Log(hr) | S1, S3→ 2.661 − 1.679(Clay) − 0.094(Silt) + 2.013(Clay^{2})S2→ −1.420(Clay) + 1.426(Clay^{2}) − 0.400(BD)(Silt)S4→ 2.090 + 0.023(SA) + 3.378(Silt^{2}) + 0.176(Clay^{2}) − 5.061(Clay)(Silt) − 0.032(Silt)(SA)S5→ 2.775 − 1.420(Clay) + 1.426(Clay^{2}) − 0.400(BD)(Silt)S6→ 2.666 − 0.007(SA) − 0.059(SOM^{2}) + 3.59 × 10 ^{−4}(SA^{2}) − 2.656(θ_{S})(Clay) + 0.610(Clay)(SOM) − 0.016(Silt)(SA) | |

θs | S1, S4→ 0.455 + 1.322(Clay)(Silt)S2→ 0.792 − 0.136(BD^{2}) + 0.093(Clay^{2})S3→ 0.311 + 1.296(Silt) − 1.818(Silt^{2}) + 0.175(Clay)(SOM)S5→ 0.811 + 0.036(SOM) − 0.163(BD^{2}) − 0.020(Clay)(SOM) | |

m | S1, S3, S4→ 0.231 + 3.236(Silt) + 2.688(Clay^{2}) − 10.96(Clay)(Silt)S2, S5→ 0.584 − 2.013(Clay) + 1.452(BD)(Silt) + 3.685(Clay^{2}) − 4.835(Clay)(Silt)S6→ 2.133 − 3.130(θ_{S}) + 0.453(Silt)(SOM) | |

FX-PDI | Log(α) | S1, S2, S4→ −1.071 − 5.543(Silt) − 3.691(Clay^{2}) + 14.613(Clay)(Silt)S3→ 1.111 − 5.497(Silt) − 3.501(Clay^{2}) + 0.015(SOM^{2}) + 14.331(Clay)(Silt)S5→ −1.128 − 5.638(Silt) − 3.652(Clay^{2}) + 0.039(BD)(SOM) + 14.814(Clay)(Silt)S6→ −0.618 − 0.391(BD^{2}) − 2.088(Silt) + 1.788(Clay^{2}) + 0.113(SOM^{2}) + 0.071(BD)(SOM) − 0.629(θ_{S})(SOM) |

Log(n) | S1, S2, S4, S5→ 0.471 − 1.489(Silt) + 3.278(Clay)(Silt)S3→ 0.526 − 1.219(Silt) − 0.011(SOM) + 2.456(Clay)(Silt) + 0.189(Clay)(SOM) S6→ 0.343 − 0.043(BD)(SOM) + 1.399(θ_{S})(Clay) − 1.192(θ_{S})(Silt) | |

θr | S1, S3→ 0.023 + 0.210(Silt) − 0.610(Silt^{2}) + 1.125(Clay^{2})S2, S5→ 0.641 − 0.226(BD^{2}) − 0.016(Silt^{2}) − 0.282(BD)(Silt)S4→ 0.060 + 1.015(Clay)(SA) − 0.007(Silt)(SA)S6→ − 0.546 + 1.363(θ_{S}) − 0.605(θ_{S})(Silt) − 0.098(Clay)(SOM) + 0.005(Clay)(SA) | |

θs | S1, S4→ 0.453 + 1.315(Clay)(Silt)S2→ 0.786 − 0.135(BD^{2}) + 0.098(Clay^{2})S3→ 0.309 + 1.299(Silt) − 1.831(Silt^{2}) + 0.174(Clay)(SOM)S5→ 0.819 − 0.171(BD^{2}) + 0.026(BD)(SOM) − 0.007(Clay)(SOM) | |

m | S1→ 1.179 − 3.118(Clay) + 3.144(Clay^{2})S2→ −0.158 + 0.574(BD)S3→ 0.791 − 0.605(Clay)(SOM)S4→ 0.931 − 1.199(Clay) + 2.20 × 10 ^{−5}(SA^{2})S5→ 0.922 − 0.012(SA) + 1.23 × 10 ^{−4}(SA^{2}) − 0.442(Clay)(SOM)S6→ 1.824 − 1.461(BD)(θ_{S}) − 0.562(Clay)(SOM) | |

FX-b-PDI | Log(α_{1}) | S1→ −1.926 − 2.231(Silt^{2}) + 2.225(Clay^{2})S2→ −1.609 + 1.772(Clay^{2}) − 1.275(BD)(Silt)S3→ −1.240 − 0.808(SOM) − 2.723(Silt^{2}) + 0.102(SOM^{2}) + 1.103(Clay)(SOM)S4→ −1.939 + 0.001(SA) − 2.198(Silt^{2}) + 2.117(Clay^{2})S5→ −1.140 + 0.016(SOM) −0.022(SOM^{2}) −1.806(BD)(Silt)S6→ −1.470 − 0.048(SOM) − 0.009(SOM^{2}) − 1.896(BD)(Silt) + 0.116(θ_{S})(SOM) + 3.515(Clay)(Silt) − 0.001(SOM)(SA) |

Log(n_{1}) | S1, S4→ 0.389 + 1.006(Clay)(Silt)S2, S5→ 0.587 − 0.083(BD^{2}) + 0.346(Clay)(Silt)S3→ 0.375 + 0.478(Clay)(Silt) + 0.206(Clay)(SOM)S6→ 0.310 + 0.385(θ_{s}^{2}) + 0.445(Clay)(Silt) | |

θr | S1→ −0.020 + 1.849(Clay)(Silt)S2→ 0.365 − 0.180(BD^{2}) + 0.217(BD)(Clay)S3→ −0.012 + 1.811(Clay)(Silt)S4→ 0.020 + 0.456(Clay)(Silt) + 0.008(Clay)(SA)S5→ 0.421 − 0.177(BD^{2}) − 0.023(BD)(SOM) + 0.002(BD)(SA) + 1.25 × 10^{−4}(SOM)(SA)S6→ 0.163 − 0.184(BD^{2}) + 0.382(BD)(θ_{s}) + 0.005(Clay)(SA) − 0.079(θ_{S})(SOM) + 0.014(BD)(SOM) | |

θs | S1→ 0.454 + 1.316(Clay)(Silt)S2→ 0.788 − 0.135(BD^{2}) + 0.096(Clay^{2})S3→ 0.313 + 1.343(Silt) − 1.871(Silt^{2}) + 0.159(Clay)(SOM)S4→ 0.454 + 1.316(Clay)(Silt)S5→ 0.779 − 0.157(BD^{2}) + 0.029(BD)(SOM) − 0.022(Clay)(SOM) + 0.002(Silt)(SA) | |

Log(α_{2}) | S1→ −0.988 − 8.727(Silt) + 8.774(Silt^{2}) + 0.324(Clay^{2}) + 7.864(Clay)(Silt)S2→ −2.414 + 0.448(BD) − 2.047(BD)(Silt) + 6.947(Clay)(Silt)S3→ −1.271 − 6.954(Silt) + 6.438(Silt^{2}) + 8.284(Clay)(Silt)S4→ −1.190 − 5.734(Silt) + 5.603(Silt^{2}) − 1.41 × 10 ^{−4}(SA^{2}) + 0.072(Silt)(SA)S5→ −0.334 − 1.015(BD) − 0.438(SOM) + 0.006(SOM)(SA)S6→ −0.460 + 1.843(θ_{s}) − 3.923(BD)(θ_{s}) + 0.008(BD)(SA) | |

Log(n_{2}) | S1→ 0.422 + 0.396(Clay^{2})S2→ 1.058 − 0.396(BD) − 0.272(Clay)S3→ 0.400 + 0.541(Clay^{2})S4→ 0.491 − 0.002(Silt)(SA)S5→ 1.314 − 0.538(BD) − 0.218(BD)(Clay) − 0.010(Silt)(SA)S6→ 0.526 − 0.287(θ_{s}^{2}) + 0.136(BD)(Clay) | |

w_{2} | S1→ 0.186 + 2.909(Clay) + 3.093(Silt^{2}) − 9.554(Clay)(Silt)S2→ 0.651 − 0.125(BD) − 0.017(Silt)S3→ 0.887 − 1.552(Silt) − 0.183(SOM) + 0.787(Silt)(SOM)S4→ 0.453 + 0.006(SA) + 0.264(Silt^{2}) + 0.905(Clay)(Silt) − 0.029(Silt)(SA)S5→ 0.590 − 0.623(Silt) − 0.051(SOM) + 0.407(Silt)(SOM)S6→ 0.510 + 0.657(Clay)(Silt) + 0.009(Silt)(SA) | |

m_{1} | S1, S4→ 0.980 − 1.234(Clay)S2, S5, S6→ −0.319 + 0.711(BD)S3→ 1.059 − 1.471(Clay) | |

m_{2} | S1, S2→ 0.617 − 0.061(Silt)S3→ 0.631 − 0.117(Silt)(SOM)S4→ 0.894 − 0.022(SA) + 2.88 × 10^{−4}(SA^{2})S5→ 0.647 − 0.129(Silt)(SOM)S6→ 0.868 − 1.471(θ_{s})(Silt) | |

K | Log(hm) | S1, S3, S4→ 1.731 + 3.860(Silt) + 1.527(Clay^{2}) − 7.986(Clay)(Silt)S2→ 2.289 + 4.692(Silt^{2}) + 0.691(Clay^{2}) − 5.440(Clay)(Silt)S5→ 1.275 + 0.674(BD) + 0.573(Clay^{2}) + 3.978(Silt^{2}) − 2.841(Clay)(Silt)S6→ 1.110 + 1.202(θ_{S}) − 0.048(SOM) − 0.395(Silt^{2}) + 1.269(Clay^{2}) + 2.803(BD)(Silt) − 6.360(Clay)(Silt) |

σ | S1, S2, S4→ 1.767 − 1.021(Clay) − 0.814(Silt) + 5.650(Clay)(Silt)S3→ 1.520 − 0.015(Silt) + 1.929(Clay)(Silt) + 0.105(Clay)(SOM)S5→ 1.537 − 0.185(BD)(Silt) + 2.142(Clay)(Silt) + 0.180(Silt)(SOM)S6→ 0.473 + 0.714(Clay) + 0.416(Silt^{2}) + 1.393(BD)(θ_{S}) | |

θr | S1, S4→ −0.002 + 0.397(Clay) + 0.366(Silt) − 0.797(Silt^{2})S2→ 0.418 − 0.139(BD^{2}) − 0.367(Silt^{2})S3→ −0.019 + 0.421(Clay) + 0.330(Silt) − 0.766(Silt^{2}) + 0.009(SOM^{2})S5→ 0.285 − 0.106(BD^{2}) − 0.356(Silt^{2}) + 0.008(SOM^{2}) + 0.164(BD)(Clay)S6→ −0.019 − 0.134(BD^{2}) − 0.526(Silt^{2}) + 0.570(BD)(θ_{S}) + 0.333(Clay)(Silt) | |

θs | S1→ 0.454 + 1.418(Clay)(Silt)S2→ 0.800 − 0.139(BD^{2}) + 0.138(Clay^{2})S3→ 0.307 + 1.368(Silt) − 1.956(Silt^{2}) + 0.187(Clay)(SOM)S4→ 0.454 + 1.418(Clay)(Silt)S5→ 0.838 − 0.175(BD^{2}) + 0.022(BD)(SOM) + 0.009(Clay)(SOM) | |

K-PDI | Log(hm) | S1, S3, S4, S5→ 1.520 + 4.933(Silt) + 2.512(Clay^{2}) − 12.216(Clay)(Silt)S2→ 1.644 + 0.389(BD) + 5.851(Silt^{2}) + 1.656(Clay2) − 8.069(Clay)(Silt)S6→ 2.046 + 0.805(θ_{S}^{2}) − 0.001(SOM) + 6.834(Silt^{2}) + 2.045(Clay^{2}) − 11.538(Clay)(Silt) |

σ | S1, S4→ 1.092 + 1.998(Silt) − 4.733(Clay)(Silt)S2→ 0.824 + 1.809(Silt) + 0.131(BD^{2}) − 3.559(Clay)(Silt)S3→ 0.973 + 1.738(Silt) + 0.070(SOM^{2}) − 4.104(Clay)(Silt)S5→ 0.448 + 1.915(Silt) + 0.220(BD^{2}) + 0.097(SOM^{2}) − 2.226(Clay)(Silt) − 0.308(Silt)(SOM)S6→ 0.847 + 1.353(BD)(Silt) + 1.178(θ_{S})(Silt) − 5.227(Clay)(Silt) + 0.005(SOM)(SA) | |

θr | S1, S4→ −0.007 + 0.692(Clay) + 0.507(Silt) − 1.087(Silt^{2})S2→ 0.337 − 0.088(BD^{2}) − 0.930(Silt^{2}) + 1.604(Clay)(Silt)S3→ −0.019 + 0.708(Clay) + 0.483(Silt) − 1.066(Silt^{2}) + 0.006(SOM^{2})S5→ 0.305 + 0.054(SOM) − 0.093(BD^{2}) − 1.135(Silt^{2}) + 2.046(Clay)(Silt) − 0.128(Clay)(SOM)S6→ −0.362 + 1.084(θ_{S}) − 0.477(Silt^{2}) + 0.003(Clay)(SA) | |

θs | S1, S4→ 0.456 + 1.315(Clay)(Silt)S2→ 0.784 − 0.133(BD^{2}) + 0.108(Clay^{2})S3→ 0.360 + 0.658(Silt) − 0.808(Silt^{2}) + 0.641(Clay^{2}) + 0.013(SOM^{2})S5→ 0.457 + 0.219(SOM) + 0.240(BD)(Clay) − 0.159(BD)(SOM) + 0.009(Silt)(SOM) + 3.01 × 10 ^{−4}(SOM)(SA) | |

K-b | Log(hm_{1}) | S1, S2, S3→ 2.582 + 0.456(Clay^{2})S4, S5→ 2.750 − 0.008(Clay)(SA)S6→ 4.154 − 1.922(BD)(θ_{S}) − 0.010(Clay)(SA) |

σ_{1} | S1, S2, S3, S4, S5, S6→ 1.356 + 0.635(Silt) | |

θr | S1, S4→ 0.057 − 0.161(Clay)(Silt)S2→ 0.160 − 0.058(BD) − 0.054(Silt^{2}) − 0.107(BD)(Clay)S3→ 0.044 + 0.005(SOM^{2}) − 0.128(Clay)(Silt)S5→ 0.184 − 0.090(BD) − 0.126(Clay) + 0.026(Silt)(SOM)S6→ −0.025 + 0.268(θ_{S}^{2}) − 0.001(Silt^{2}) + 0.032(BD)(Clay) + 0.220(θ_{S})(Silt) − 0.799(Clay)(Silt) | |

θs | S1, S4→ 0.455 + 1.330(Clay)(Silt)S2→ 0.792 − 0.136(BD^{2}) + 0.098(Clay^{2})S3→ 0.317 + 1.269(Silt) − 1.786(Silt^{2}) + 0.176(Clay)(SOM)S5→ 0.813 + 0.035(SOM) − 0.164(BD^{2}) − 0.019(Clay)(SOM) | |

Log(hm_{2}) | S1→ 3.348 − 4.328(Silt) + 5.114(Silt^{2}) + 2.456(Clay)(Silt)S2→ 2.229 − 5.031(Clay) − 0.415(Silt) + 6.065(BD)(Clay)S3→ 2.631 − 0.516(Silt^{2}) + 0.524(Clay)(SOM) + 0.023(Silt)(SOM)S4→ 2.766 + 2.57 × 10 ^{−4}(SA^{2}) − 0.875(Clay)S5→ 2.480 + 0.009(SOM)(SA)S6→ 2.470 − 1.440(θ_{S}^{2}) − 0.002(SOM)(SA) + 0.521(θ_{S})(SOM) + 0.018(SA) | |

σ_{2} | S1→ 1.417 + 1.936(Clay^{2})S2→ 0.944 − 0.990(Clay^{2}) + 2.205(BD)(Clay)S3→ 1.304 + 0.953(Clay)(SOM)S4→ 1.416 + 1.936(Clay^{2})S5, S6→ 1.306 + 0.010(SOM)(SA) | |

w_{2} | S1, S2, S3, S4→ 0.495 + 0.030(Clay)S5→ 0.486 − 0.079(Clay) + 0.001(BD)(SA)S6→ 0.607 + 0.215(Clay) + 0.009(Silt^{2}) − 0.264(BD)(Clay) − 0.188(BD)(Silt) | |

K-b-PDI | Log(hm_{1}) | S1, S2, S3, S4→ 2.346 − 0.715(Clay^{2})S5→ 2.328 − 0.812(Clay^{2}) + 0.001(SOM)(SA)S6→ 2.219 − 2.759(Clay^{2}) + 1.885(θ_{S})(Clay) + 2.99 × 10^{−4}(SOM)(SA) |

σ_{1} | S1, S2, S3, S4, S5→ 1.107 + 1.719(Clay)(Silt)S6→ 1.501 − 3.464(Clay) − 5.711(θ_{S})(Silt) + 0.516(θ_{S})(SOM) + 15.642(Clay)(Silt) | |

θr | S1, S3, S4→ 0.180 − 0.546(Silt) + 2.256(Clay)(Silt)S2, S5→ 0.929 − 0.534(BD) − 0.220(Silt^{2})S6→ −0.342 + 0.933(θ_{S}) − 0.001(SOM) − 0.012(SOM^{2}) + 0.514(θ_{S})(Clay) − 0.460(Clay)(Silt) | |

θs | S1, S4→ 0.455 + 1.315(Clay)(Silt)S2→ 0.787 − 0.134(BD^{2}) + 0.100(Clay^{2})S3→ 0.312 + 1.293(Silt) − 1.822(Silt^{2}) + 0.173(Clay)(SOM)S5→ 0.808 + 0.034(SOM) − 0.161(BD^{2}) − 0.017(Clay)(SOM) | |

Log(hm_{2}) | S1, S2→ 2.499 − 1.642(Clay^{2})S3→ 2.367 + 0.240(SOM) − 1.705(Clay^{2}) − 0.080(SOM^{2})S4→ 2.422 − 0.012(Clay)(SA)S5→ 2.265 − 0.063(SOM^{2}) + 0.163(BD)(SOM) − 0.010(Clay)(SA)S6→ 1.038 + 1.894(BD)(θ_{S}) − 0.010(Clay)(SA) | |

σ_{2} | S1, S2, S3→ 1.247 + 0.343(Clay)(Silt)S4, S5→ 1.327 + 0.003(SA) − 1.469(Silt^{2})S6→ −0.448 + 2.731(BD)(θ_{S}) − 0.319(θ_{S})(SOM) | |

w_{2} | S1→ 0.534 + 0.077(Clay) − 0.410(Clay)(Silt)S2→ 0.618 − 0.055(BD) − 0.099(Clay)S3→ 0.342 + 0.245(SOM) − 0.065(SOM^{2})S4→ 0.537 - 0.194(Clay)(Silt)S5→ 0.492 + 0.116(Clay) + 0.029(SOM) − 0.480(Clay)(Silt)S6→ 0.337 + 0.130(SOM) - 0.059(SOM^{2}) + 0.078(BD)(SOM) + 0.002(Silt)(SA) | |

VG | α | S1, S2, S4→ −1.186 − 4.644(Silt) − 3.038(Clay^{2}) + 12.213(Clay)(Silt)S3→ −1.241 − 4.723(Silt) + 0.055(SOM) − 3.005(Clay^{2}) + 12.287(Clay)(Silt)S5→ −1.462 − 3.087(Silt) + 0.080(SOM) − 0.019(BD)(SOM) + 5.604(Clay)(Silt)S6→ −1.344 − 0.469(θ_{S}^{2}) − 2.131(BD)(Silt) + 0.061(BD)(SOM) + 4.481(Clay)(Silt) |

n | S1→ 0.216 − 0.100(Silt) − 0.109(Clay)(Silt)S2→ 0.279 − 0.030(BD^{2}) − 0.040(BD)(Silt) − 0.417(Clay)(Silt)S3→ 0.222 − 0.088(Silt) − 0.007(SOM) − 0.127(Clay)(Silt)S4→ 0.192 + 0.109(Clay) − 0.020(Silt) − 0.472(Clay)(Silt)S5→ 0.221 − 0.059(BD)(Silt) − 0.172(Clay)(Silt) − 0.017(Silt)(SOM)S6→ 0.443 − 0.320(BD)(θ_{S}) − 0.048(BD)(Silt) − 0.238(Clay)(Silt) + 0.009(Silt)(SOM) | |

θr | S1→ 0.016 + 0.411(Clay) − 0.372(Silt^{2})S2→ 0.296 + 0.115(Clay) − 0.111(BD^{2}) − 0.440(Silt^{2})S3→ 0.043 − 0.939(Silt^{2}) + 0.015(SOM^{2}) + 1.691(Clay)(Silt) − 0.088(Clay)(SOM)S4→ 0.017 + 0.411(Clay) − 0.013(Silt) − 0.354(Silt^{2})S5→ 0.125 + 0.308(Clay) + 0.029(SOM^{2}) − 0.174(BD)(Silt) − 0.068(BD)(SOM)S6→ −0.100 − 0.385(Silt^{2}) + 0.174(BD)(θ_{S}) + 0.624(θ_{S})(Clay) | |

θs | S1, S4→ 0.456 + 1.357(Clay)(Silt)S2→ 0.781 − 0.131(BD^{2}) + 0.143(Clay^{2})S3→ 0.335 + 1.182(Silt) − 1.706(Silt^{2}) + 0.183(Clay)(SOM)S5→ 0.815 + 0.033(SOM) − 0.163(BD^{2}) − 0.009(Clay)(SOM) | |

VG-PDI | α | S1, S2, S4→ −1.241 − 4.335(Silt) − 2.919(Clay^{2}) + 11.276(Clay)(Silt)S3→ −1.509 − 2.870(Silt) + 0.031(SOM^{2}) + 5.128(Clay)(Silt)S5→ −1.545 − 2.947(Silt) + 0.064(BD)(SOM) + 5.326(Clay)(Silt)S6→ −1.442 + 0.013(BD^{2}) + 0.088(SOM^{2}) − 1.826(BD)(Silt) − 0.015(BD)(SOM) − 0.286(θ_{S})(SOM) + 3.633(Clay)(Silt) |

n | S1, S2, S4→ 0.311 − 0.556(Silt) + 1.207(Clay)(Silt)S3, S5→ 0.340 − 0.501(Silt) − 0.031(SOM) + 1.125(Clay)(Silt)S6→ 0.390 − 0.140(θ_{S}^{2}) − 0.369(BD)(Silt) − 0.021(BD)(SOM) + 0.922(Clay)(Silt) | |

θr | S1, S4→ 0.018 + 0.875(Clay) − 0.029(Silt) − 0.539(Silt^{2})S2→ 0.260 + 0.613(Clay) − 0.098(BD^{2}) − 0.640(Silt^{2})S3→ 0.016 + 0.840(Clay) − 0.539(Silt^{2}) + 0.032(Clay)(SOM) − 0.021(Silt)(SOM)S5→ 0.256 + 0.618(Clay) − 0.100(BD^{2}) − 0.653(Silt^{2}) + 0.005(BD)(SOM)S6→ −0.100 − 0.140(BD^{2}) − 1.342(Silt^{2}) + 0.716(BD)(θ_{S}) + 1.828(Clay)(Silt) | |

θs | S1, S4→ 0.454 + 1.328(Clay)(Silt)S2→ 0.787 − 0.135(BD^{2}) + 0.108(Clay^{2})S3→ 0.312 + 1.289(Silt) − 1.834(Silt^{2}) + 0.178(Clay)(SOM)S5→ 0.746 − 0.141(BD^{2}) + 0.173(Clay^{2}) + 0.027(BD)(SOM) | |

VG-b | Log(α_{1}) | S1, S2, S4→ −2.389 + 0.456(Silt^{2})S3→ −2.225 − 0.062(SOM^{2})S5→ −2.218 − 0.083(BD)(SOM)S6→ − 0.015 − 3.238(BD)(θ_{S}) |

Log(n_{1}) | S1, S2, S4→ 0.378 − 0.676(Clay)(Silt)S3, S5→ 0.410 + 0.008(SOM) − 0.017(SOM^{2}) − 0.772(Clay)(Silt)S6→ 0.643 − 0.588(θ_{S}) − 0.034(SOM^{2}) + 0.131(θ_{S})(SOM) − 0.170(Clay)(Silt) | |

θr | S1, S4→ 0.003 + 0.119(Silt^{2})S2, S5→ 0.015-0.025(Silt^{2}) − 0.057(BD)(Clay) + 0.066(BD)(Silt)S3→ 0.002 + 0.118(Silt^{2}) + 4.51 × 10^{−4}(SOM^{2})S6→ 0.002 + 0.227(Silt) + 0.065(Silt^{2}) − 0.346(θ_{S})(Silt) | |

θs | S1, S4→ 0.456 + 1.320(Clay)(Silt)S2→ 0.780 − 0.130(BD^{2}) + 0.115(Clay^{2})S3→ 0.327 + 1.193(Silt) − 1.670(Silt^{2}) + 0.177(Clay)(SOM)S5→ 0.803 + 0.034(SOM) − 0.159(BD^{2}) − 0.011(Clay)(SOM) | |

Log(α_{2}) | S1, S3→ −1.298–3.945(Clay) − 5.678(Silt^{2}) + 9.879(Clay)(Silt)S2→ −1.316 − 0.070(BD^{2}) − 2.675(Silt^{2}) − 1.388(BD)(Clay) + 0.583(Clay)(Silt)S4→ −2.135 − 0.006(Silt)(SA)S5→ −2.246 + 0.002(BD)(SA)S6→ −1.235 − 1.756(BD)(θ_{S}) + 0.205(BD)(SOM) | |

Log(n_{2}) | S1, S2, S3, S4, S5, S6→ 0.228 + 0.367(Silt) | |

w_{2} | S1, S2, S3, S4, S5, S6→ 0.496 − 0.220(Silt) + 0.582(Silt^{2}) | |

VG-b-PDI | Log(α_{1}) | S1, S4→ −1.576 − 4.139(Clay) − 4.883(Silt^{2}) + 13.717(Clay)(Silt)S2→ −1.908 − 0.443(BD)(Silt) + 0.250(Clay)(Silt)S3→ −2.014 + 0.236(Clay^{2}) − 0.179(Silt)(SOM)S5→ −1.844 − 0.086(BD^{2}) − 0.191(Silt)(SOM)S6→ −1.064 − 0.311(BD^{2}) − 1.289(θ_{S}^{2}) − 0.105(Silt)(SOM) |

Log(n_{1}) | S1→ 0.502 − 0.080(Silt) + 0.025(Clay^{2}) − 1.172(Clay)(Silt)S2→ 0.697 − 1.143(Silt) + 1.216(Silt^{2}) − 0.283(BD)(Clay)S3→ 0.487 + 0.230(Silt^{2}) + 0.389(Clay^{2}) − 1.974(Clay)(Silt)S4→ 0.448 − 0.003(SA) − 0.425(Silt^{2}) + 4.06 × 10^{−5}(SA^{2})S5→ 0.361 − 0.297(Silt^{2}) − 3.57 × 10^{−5}(SA^{2}) + 0.002(BD)(SA)S6→ 0.320 − 0.310(Silt^{2}) − 3.35 × 10 ^{−5}(SA^{2}) + 0.062(BD)(θ_{S}) + 0.002(BD)(SA) | |

θr | S1, S3, S4→ 0.025 + 1.200(Clay)(Silt)S2→ 0.501 − 0.262(BD) + 0.631(Silt^{2}) − 0.243(BD)(Silt)S5→ 0.203 + 1.532(Silt^{2}) + 2.97 × 10^{−5}(SA^{2}) − 0.651(BD)(Silt)S6→ −0.195 + 0.590(θ_{S}) − 0.317(Silt) + 0.660(Silt^{2}) − 0.001(SOM^{2}) + 2.19 × 10^{−5}(SA^{2}) | |

θs | S1, S4→ 0.454 + 1.321(Clay)(Silt)S2→ 0.787 − 0.135(BD^{2}) + 0.100(Clay^{2})S3→ 0.310 + 1.301(Silt) − 1.828(Silt^{2}) + 0.173(Clay)(SOM)S5→ 0.773 + 0.041(SOM) − 0.150(BD^{2}) + 5.26 × 10^{−6}(SA^{2}) − 0.040(Clay)(SOM) + 0.001(Silt)(SA) | |

Log(α_{2}) | S1, S3→ −2.307 + 2.306(Clay^{2})S2→ −2.012 − 0.141(BD^{2}) + 1.759(Clay^{2})S4→ −2.278 − 1.677(Clay) + 0.017(SA) + 6.323(Clay^{2}) − 0.042(Clay)(SA)S5→ −2.269 + 0.008(BD)(SA)S6→ 0.685 − 0.327(BD^{2}) − 3.022(BD)(θ_{S}) + 1.94 × 10^{−4}(SOM)(SA) | |

Log(n_{2}) | S1, S3→ 0.336 − 0.486(Clay)(Silt)S2→ −0.057 + 0.175(BD^{2}) + 0.180(BD)(Clay)S4→ 0.343 − 0.006(Silt)(SA)S5→ −0.726 + 0.680(BD) + 1.658(Clay)(Silt)S6→ 0.332 + 0.006(SA) − 0.019(θ_{S})(SA) + 0.015(Silt)(SA) | |

w_{2} | S1, S3, S4→ 0.673 − 1.012(Clay)(Silt)S2, S5, S6→ 0.722 − 0.020(BD^{2}) − 1.174(Clay)(Silt) | |

VG_{m} | Log(α) | S1, S2→ −1.803 − 10.338(Silt^{2}) − 4.471(Clay^{2}) + 17.146(Clay)(Silt)S3→ −0.728 − 7.770(Silt) − 5.066(Clay^{2}) + 0.019(SOM^{2}) + 20.084(Clay)(Silt)S4→ −0.680 − 7.826(Silt) − 5.296(Clay^{2}) + 20.425(Clay)(Silt)S5→ −1.597 + 0.036(SOM) − 0.111(BD2) − 10.071(Silt^{2}) − 4.335(Clay^{2}) + 15.972(Clay)(Silt)S6→ −1.488 + 3.123(Clay) + 0.021(SOM^{2}) − 2.398(BD)(Silt) − 1.847(θ_{S})(Clay) |

Log(n) | S1, S2, S3, S4, S5→ 0.611 − 1.993(Silt) + 4.257(Clay)(Silt)S6→ 0.618 − 2.008(Silt) + 4.349(Clay)(Silt) − 0.032(Clay)(SOM) | |

θr | S1, S3→ 0.025 + 0.185(Silt) − 0.386(Silt^{2}) + 0.355(Clay^{2})S2→ 0.436 − 0.249(BD) + 0.002(Silt^{2}) − 0.099(BD)(Silt)S4→ 0.018 + 0.274(Silt) − 0.500(Silt^{2}) + 0.003(Clay)(SA)S5→ 0.445 − 0.271(BD) − 0.190(Silt^{2})S6→ −0.306 + 0.777(θ_{S}) + 1.99 × 10^{−5}(SA^{2}) − 0.184(θ_{S})(Silt) − 0.025(Clay)(SOM) − 0.005(Silt)(SA) | |

θs | S1, S4→ 0.457 + 1.303(Clay)(Silt)S2→ 0.778 − 0.129(BD^{2}) + 0.116(Clay^{2})S3→ 0.341 + 1.112(Silt) − 1.571(Silt^{2}) + 0.177(Clay)(SOM)S5→ 0.808 + 0.039(SOM) − 0.162(BD^{2}) − 0.025(Clay)(SOM) | |

Log(m) | S1, S4→ −0.394 − 4.345(Clay) + 1.782(Silt) + 3.577(Clay^{2})S2→ −0.648 − 1.738(BD)(Clay) + 1.337(BD)(Silt)S3→ −0.393 − 4.326(Clay) + 1.782(Silt) + 3.543(Clay^{2}) − 0.001(SOM^{2})S5, S6→ −0.635 − 0.005(SOM^{2}) − 1.750(BD)(Clay) + 1.342(BD)(Silt) | |

VG_{m}-PDI | Log(α) | S1, S4→ −1.927 − 8.150(Silt^{2}) − 3.484(Clay^{2}) + 14.055(Clay)(Silt)S2→ −1.819 − 10.975(Silt^{2}) − 4.945(Clay^{2}) + 16.931(Clay)(Silt)S3→ −2.063 − 7.678(Silt^{2}) − 2.788(Clay^{2}) + 0.035(SOM^{2}) + 13.143(Clay)(Silt)S5, S6→ −1.900 − 10.393(Silt^{2}) − 4.004(Clay^{2}) + 0.040(SOM^{2}) + 15.146(Clay)(Silt) |

Log(n) | S1→ 0.089 + 1.372(Clay) − 1.150(Silt^{2})S2→ 0.563 + 0.586(Clay) − 1.228(BD)(Silt)S3→ 0.217 − 2.416(Silt^{2}) + 0.009(SOM^{2}) + 4.377(Clay)(Silt) − 0.035(Silt)(SOM)S4→ 0.194 + 0.016(SA) − 0.026(Silt)(SA)S5→ 0.564 − 1.097(BD)(Silt) + 0.004(SOM)(SA)S6→ −0.191 + 2.172(θ_{S}) − 0.009(BD)(SA) − 4.590(θ_{S})(Silt) + 3.475(Clay)(Silt) − 0.559(Clay)(SOM) + 0.009(SOM)(SA) | |

θr | S1, S3, S4→ 0.152 − 0.557(Silt^{2}) + 1.119(Clay^{2})S2, S5→ 0.243 − 0.705(Silt^{2}) + 0.617(Clay^{2})S6→ −0.359 − 2.393(Silt^{2}) + 1.391(BD)(θ_{S}) − 0.628(BD)(Clay) + 3.541(Clay)(Silt) | |

θs | S1, S4→ 0.552 − 0.313(Silt) + 1.347(Clay)(Silt)S2→ 0.740 + 0.223(Silt) − 0.268(BD^{2}) − 0.276(Clay^{2})S3→ 0.438 + 1.279(Clay)(Silt) + 0.046(Silt)(SOM)S5→ 0.750 + 0.095(SOM) − 0.279(BD^{2}) − 0.205(Clay)(SOM) | |

Log(m) | S1→ 0.062 − 5.386(Silt) + 9.343(Silt^{2})S2→ −0.877 + 0.486(BD^{2})S3→ −0.622 + 0.614(SOM) − 0.226(SOM^{2}) − 2.088(Clay^{2})S4→ −0.328 − 0.352(Clay) + 6.84 × 10^{−5}(SA^{2}) − 0.021(Clay)(SA)S5→ −0.849 + 0.483(BD^{2}) − 0.144(SOM^{2}) + 0.705(Silt)(SOM)S6→ −0.146 − 1.321(θ_{S})(SOM) + 1.542(Silt)(SOM) | |

VG_{m}-b | Log(α_{1}) | S1, S2, S3→ −1.356 − 6.170(Silt) − 7.060(Clay^{2}) + 17.793(Clay)(Silt)S4→ −2.806 + 0.037(SA) − 0.053(Clay)(SA)S5→ −2.536-0.011(BD)(SA)S6→ −1.834 − 1.551(BD)(θ_{S}) |

Log(n_{1}) | S1→ 0.237 + 2.284(Silt^{2}) − 0.922(Clay^{2})S2→ −0.319 + 2.216(BD)(Silt)S3→ −0.005 + 1.693(Silt) − 0.447(Clay)(SOM)S4→ 0.237 + 2.284(Silt^{2}) − 0.922(Clay^{2})S5→ 0.097 + 0.499(BD)(Silt) + 0.499(Silt)(SOM)S6→ −0.440 + 0.753(θ_{S}) + 1.259(BD)(Silt) | |

θr | S1, S4→ 0.019 + 0.859(Silt^{2}) − 0.285(Clay^{2})S2→ 0.120 + 2.141(Silt^{2}) − 1.078(Clay^{2}) + 0.533(BD)(Clay) − 0.968(BD)(Silt)S3→ 0.001 + 0.835(Silt^{2}) − 0.257(Clay^{2}) + 0.009(SOM^{2})S5→ −0.010 − 0.260(Silt^{2}) − 0.114(Clay^{2}) + 0.154(BD)(Silt) + 3.57 × 10^{−4}(SOM)(SA)S6→ 0.027 − 0.177(Clay^{2}) + 0.031(θ_{S})(Silt) + 3.06 × 10^{−4}(SOM)(SA) | |

θs | S1, S4→ 0.378 + 0.281(Silt) + 0.463(Clay)(Silt)S2→ 0.743 + 0.215(Silt) − 0.267(BD^{2}) − 0.282(Clay^{2})S3→ 0.396 + 0.582(Clay)(Silt) + 0.147(Silt)(SOM)S5→ 0.836 + 0.050(SOM) − 0.183(BD^{2}) − 0.057(Clay)(SOM) | |

Log(α_{2}) | S1→ −1.936 − 2.545(Silt^{2}) + 2.001(Clay)(Silt)S2→ −1.271 − 0.517(BD) − 2.025(Silt^{2})S3→ −2.122 − 0.053(SOM^{2}) + 0.628(Clay)(SOM)S4→ −2.448 + 0.032(SA) + −3.84 × 10^{−4}(SA^{2})S5→ −2.161 − 0.086(SOM^{2}) − 0.016(Clay)(SA) + 0.010(SOM)(SA)S6→ −1.126 − 1.653(BD)(θ_{s}) + 0.003(SOM)(SA) | |

Log(n_{2}) | S1→ 0.409 + 0.149(Clay) − 0.096(Silt)S2→ 2.146 − 1.477(BD) − 2.658(Clay)(Silt)S3→ 0.302 + 0.161(Clay) − 0.084(SOM) + 0.469(Silt)(SOM)S4→ 0.307 + 0.051(Clay) − 1.79 × 10 ^{−4}(SA^{2}) + 0.037(Silt)(SA)S5→ 0.938 − 0.423(BD) − 0.560(Clay)(Silt)S6→ 0.154 + 0.559(θ_{s}^{2}) | |

w_{2} | S1, S2, S4→ 0.391 + 0.602(Silt^{2})S3→ 0.451 − 0.062(SOM) + 0.736(Silt^{2})S5, S6→ 0.585 − 0.563(Silt^{2}) + 0.022(BD)(SOM) | |

Log(m_{1}) | S1→ −0.618 + 0.238(Clay)(Silt)S2→ −0.552 + 0.456(BD) + 5.074(Silt^{2}) − 3.362(BD)(Silt)S3→ −0.468 + 0.127(Clay) − 0.494(SOM) + 0.083(SOM^{2}) + 0.680(Silt)(SOM)S4→ −0.483 − 0.014(SA) + 0.037(Silt)(SA)S5→ 0.433 − 0.773(BD) − 9.964(Silt^{2}) + 3.258(BD)(Silt)S6→ −0.418 − 0.645(θ_{s}^{2}) + 0.025(θ_{s})(SA) − 0.014(Clay)(SA) | |

Log(m_{2}) | S1, S3→ −0.275 − 2.238(Clay)(Silt)S2→ −1.630 + 1.152(BD)S4→ −0.318 − 0.021(Silt)(SA)S5, S6→ −1.101 + 0.569(BD) | |

VG_{m}-b-PDI | Log(α_{1}) | S1 − 5→ −2.475 + 0.556(Clay)S6→ 0.635 − 4.245(BD)(θ_{s}) + 0.329(BD)(Clay) |

Log(n_{1}) | S1→ 0.226 + 3.283(Silt^{2}) − 0.629(Silt) − 0.469(Clay)(Silt)S2→ 0.125 + 2.711(Silt^{2}) − 0.222(BD)(Silt)S3, S5, S6→ 0.127 + 0.155(Clay)(Silt) + 0.457(Silt)(SOM)S4→ 0.134 + 2.418(Silt^{2}) − 0.600(Clay)(Silt) | |

θr | S1, S3, S4→ 0.061 + 0.810(Clay)(Silt)S2, S5→ 0.026 + 0.261(BD)(Clay) + 0.149(Clay)(Silt)S6→ 0.114 − 0.516(θ_{s})(Silt) + 1.199(Clay)(Silt) | |

θs | S1, S4→ 0.455 + 1.313(Clay)(Silt)S2→ 0.787 − 0.134(BD^{2}) + 0.099(Clay^{2})S3→ 0.312 + 1.294(Silt) − 1.821(Silt^{2}) + 0.172(Clay)(SOM)S5→ 0.773 + 0.042(SOM) − 0.150(BD^{2}) + 5.86 × 10 ^{−6}(SA^{2}) − 0.042(Clay)(SOM) + 0.001(Silt)(SA) | |

Log(α_{2}) | S1→ −1.571 − 2.743(Silt) − 2.793(Clay^{2}) + 8.305(Clay)(Silt)S2→ −1.675 − 1.160(Silt) − 0.050(BD^{2}) + 1.709(Clay)(Silt)S3, S4→ −1.778 − 1.233(Silt) + 2.161(Clay)(Silt)S5→ −1.861 + 0.070(BD) − 1.604(Silt) + 2.594(Clay)(Silt) + 0.171(Silt)(SOM)S6→ −1.687 − 0.465(θ_{s}^{2}) + 0.028(SOM^{2}) − 0.948(BD)(Silt) + 2.175(Clay)(Silt) | |

Log(n_{2}) | S1, S3, S4→ 0.525 − 0.494(Silt^{2})S2, S5→ 0.778 − 0.638(Silt^{2}) − 0.598(BD)(Clay)S6→ 0.412 − 0.706(Silt^{2}) + 0.514(BD)(θ_{s}) − 0.590(BD)(Clay) | |

w_{2} | S1→ 0.229 + 0.587(Silt) + 0.399(Clay^{2})S2, S5→ 0.386 + 0.209(BD)(Silt)S3, S4→ 0.267 + 0.450(Clay) + 1.009(Silt^{2}) − 0.558(Clay)(Silt)S6→ 0.583 − 0.164(BD)(θ_{s}) | |

Log(m_{1}) | S1→ −0.458 − 1.166(Silt^{2})S2→ −1.113 + 0.347(BD^{2})S3→ −0.175 − 1.331(Clay^{2}) − 0.097(Clay)(SOM) − 0.465(Silt)(SOM)S4→ −0.384 − 0.021(Silt)(SA)S5→ 0.001 − 1.293(Silt) − 0.005(BD)(SA)S6→ 0.382 − 1.509(θ_{s}) − 0.051(SOM) | |

Log(m_{2}) | S1, S3→ − 0.307 + 0.736(Clay) − 1.272(Silt)S2→ −0.401 − 1.273(Silt) + 0.868(BD)(Clay)S4→ −0.079 − 1.255(Silt) + 0.001(SA)S5→ −0.010 − 1.325(Silt) − 0.001(BD)(SA)S6→ −0.448 − 1.374(Silt) + 0.688(BD)(θ_{s}) − 0.021(BD)(SOM) − 0.001(BD)(SA) |

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**Figure 1.**Schematic illustrations of the data collection, pedotransfer function (PTF) development and SWRC estimation workflow used in this study. SHM: soil hydraulic models, SWRC: soil water retention curve.

**Figure 3.**Scatter plots of the fitted (dark blue) and the PTF-estimated (white) versus the measured soil water content data (scenario 6). 1 BC: Brooks and Corey [34], FX: Fredlund and Xing [35], K: Kosugi [36], VG and VGm: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively.

**Figure 4.**Performance of the 16 models for 4 textures and the six modeling scenarios (S1–S6). BC: Brooks and Corey [34], FX: Fredlund and Xing [35], K: Kosugi [36], VG and VG

_{m}: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively. L: Loam (1087 data pairs), CL: Clay Loam (1277 data pairs), SL: Silt Loam (1348 data pairs), C: Clay (3708 data pairs). MBE: mean bias error (cm

^{3}cm

^{−3}).

**Figure 5.**Performance of the 16 models across tension classes and the six modeling scenarios (S1–S6). BC: Brooks and Corey [34], FX: Fredlund and Xing [35], K: Kosugi [36], VG and VGm: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively. pF classes (cm of water): C1 (<100), C2 (100–200), C3 (200–300), C4 (300–400), C5 (400–500), C6 (500–600), C7 (600–700), C8 (700–800), C9 (800–900), C10 (900–1000) and C11 (>1000). MBE: mean bias error (cm

^{3}cm

^{−3}).

**Figure 6.**Mean absolute error (MAE) values for all 16 models with the fitted versus the PTF-estimated parameters (scenario 6). 1 BC: Brooks and Corey [34], FX: Fredlund and Xing [35], K: Kosugi [36], VG and VG

_{m}: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively. MBE: mean bias error (cm

^{3}cm

^{−3}).

Basic Soil Properties | HYPROP Measured Data | ||||||||
---|---|---|---|---|---|---|---|---|---|

Sand (%) | Silt (%) | Clay (%) | BD (Mg m ^{−3}) | SOM (%) | SA (%) | pF (-) | SWC (cm ^{3} cm^{−3}) | θ_{s}(m ^{3} cm^{−3}) | |

Mean | 35.20 | 30.67 | 34.13 | 1.23 | 1.18 | 30.72 | 1.77 | 0.47 | 0.59 |

SD | 17.38 | 8.68 | 15.05 | 0.18 | 0.59 | 18.91 | 0.78 | 0.12 | 0.07 |

Max. | 83.60 | 57.60 | 62.20 | 1.66 | 3.07 | 75.40 | 3.91 | 0.69 | 0.69 |

Min. | 5.90 | 5.20 | 9.40 | 0.91 | 0.01 | 1.50 | −2.00 | 0.05 | 0.40 |

_{s}: saturated water content determined by the HYPROP system; pF: the logarithmic transformation of the soil tension in cm of water; SD: standard deviation.

**Table 2.**The water retention models and their parameters estimated using parametric PTFs in this study.

Model ^{1} | Parameters | Source |
---|---|---|

BC | α, λ, θ_{r}, θ_{s} | [34] |

FX | α, n, h_{r}, θ_{s}, m | [8,30,31,32,33,35] |

FX-PDI | α, n, θ_{r}, θ_{s}, m | |

FX-b-PDI | α_{1}, n_{1}, θ_{r}, θ_{s}, α_{2}, n_{2}, w_{2}, m_{1}, m_{2} | |

K | h_{m}, σ, θ_{r}, θ_{s} | [9,30,31,32,33,36] |

K-PDI | h_{m}, σ, θ_{r}, θ_{s} | |

K-b | h_{m}_{1}, σ_{1}, θ_{r}, θ_{s}, h_{m}_{2}, σ_{2}, w_{2} | |

K-b-PDI | h_{m}_{1}, σ_{1}, θ_{r}, θ_{s}, h_{m}_{2}, σ_{2}, w_{2} | |

VG | α, n, θ_{r}, θ_{s} | [8,24,30,31,32,33] |

VG-PDI | α, n, θ_{r}, θ_{s} | |

VG-b | α_{1}, n_{1}, θ_{r}, θ_{s}, α_{2}, n_{2}, w_{2} | |

VG-b-PDI | α_{1}, n_{1}, θ_{r}, θ_{s}, α_{2}, n_{2}, w_{2} | |

VG_{m} | α, n, θ_{r}, θ_{s}, m | |

VG_{m}-PDI | α, n, θ_{r}, θ_{s}, m | |

VG_{m}-b | α_{1}, n_{1}, θ_{r}, θ_{s}, m_{1}, α_{2}, n_{2}, m_{2}, w_{2} | |

VG_{m}-b-PDI | α_{1}, n_{1}, θ_{r}, θ_{s}, m_{1}, α_{2}, n_{2}, m_{2}, w_{2} |

**Table 3.**Combinations of input attributes used and the regression terms considered to develop parametric pedotransfer functions.

Scenario | Input Parameters ^{1} | Regression Terms Considered |
---|---|---|

S1 | Silt, Clay | Silt, Clay, Silt^{2}, Clay^{2}, Clay × Silt |

S2 | Silt, Clay, BD | BD, Clay, Silt, BD^{2}, Silt^{2}, Clay^{2}, BD × Clay, BD × Silt, Clay × Silt |

S3 | Silt, Clay, SOM | Clay, Silt, SOM, Silt^{2}, Clay^{2}, SOM^{2}, Clay × Silt, Clay × SOM, Silt × SOM |

S4 | Silt, Clay, SA | Clay, Silt, SA, Silt^{2}, Clay^{2}, SA^{2}, Clay × Silt, Clay × SA, Silt × SA |

S5 | Silt, Clay, SA, BD, SOM | BD, Clay, Silt, SOM, SDA, BD^{2}, Silt^{2}, Clay^{2}, SOM^{2}, SA^{2}, BD × Clay, BD × Silt, BD × SOM, BD × SA, Clay × Silt, Clay × SOM, Clay × SA, Silt × SOM, Silt × SA, SOM × SA |

S6 | Silt, Clay, SA, BD, SOM, θs | BD, θs, Clay, Silt, OM, SA, BD^{2}, θs^{2}, Silt^{2}, Clay^{2}, SOM^{2}, SA^{2}, BD × θs, BD × Clay, BD × Silt, BD × SOM, BD × SA, θs × Clay, θs × Silt, θs × OM, θs × SA, Clay × Silt, Clay × SOM, Clay × SA, Silt × SOM, Silt × SA, SOM × SA |

^{1}BD: bulk density; SA: percentage of stable aggregates; SOM: soil organic matter content.

**Table 4.**Performance evaluation of the unimodal and unimodal-Peters–Durner–Iden (PDI) parametric PTFs for all modeling scenarios (S1–S6).

Models | S1 | S2 | S3 | S4 | S5 | S6 | |
---|---|---|---|---|---|---|---|

BC | RMSE | 0.054 | 0.051 | 0.069 | 0.054 | 0.074 | 0.037 |

MAE | 0.042 | 0.039 | 0.059 | 0.042 | 0.056 | 0.027 | |

r | 0.91 | 0.92 | 0.84 | 0.91 | 0.83 | 0.96 | |

MBE | −0.006 | −0.006 | −0.007 | −0.006 | −0.005 | −0.002 | |

FX | RMSE | 0.07 | 0.063 | 0.077 | 0.063 | 0.074 | 0.057 |

MAE | 0.052 | 0.049 | 0.062 | 0.049 | 0.053 | 0.040 | |

r | 0.85 | 0.88 | 0.81 | 0.88 | 0.86 | 0.91 | |

MBE | 0.009 | 0.007 | 0.003 | 0.007 | −0.005 | 0.007 | |

K | RMSE | 0.048 | 0.055 | 0.063 | 0.048 | 0.051 | 0.038 |

MAE | 0.039 | 0.044 | 0.053 | 0.039 | 0.041 | 0.029 | |

r | 0.92 | 0.90 | 0.86 | 0.92 | 0.91 | 0.96 | |

MBE | −0.005 | −0.006 | −0.006 | −0.005 | −0.006 | −0.012 | |

VG | RMSE | 0.051 | 0.052 | 0.064 | 0.051 | 0.056 | 0.036 |

MAE | 0.041 | 0.041 | 0.054 | 0.041 | 0.045 | 0.026 | |

r | 0.91 | 0.91 | 0.86 | 0.91 | 0.90 | 0.96 | |

MBE | −0.009 | −0.009 | −0.01 | −0.009 | −0.009 | −0.012 | |

VG_{m} | RMSE | 0.054 | 0.056 | 0.069 | 0.053 | 0.056 | 0.034 |

MAE | 0.041 | 0.043 | 0.056 | 0.042 | 0.044 | 0.024 | |

r | 0.90 | 0.90 | 0.84 | 0.91 | 0.89 | 0.96 | |

MBE | −0.003 | −0.002 | −0.004 | −0.005 | −0.001 | −0.002 | |

FX-PDI | RMSE | 0.052 | 0.065 | 0.07 | 0.055 | 0.065 | 0.042 |

MAE | 0.041 | 0.049 | 0.06 | 0.043 | 0.05 | 0.029 | |

r | 0.91 | 0.87 | 0.84 | 0.90 | 0.87 | 0.95 | |

MBE | −0.008 | −0.009 | −0.007 | −0.007 | −0.007 | −0.006 | |

K-PDI | RMSE | 0.045 | 0.049 | 0.046 | 0.045 | 0.049 | 0.038 |

MAE | 0.036 | 0.039 | 0.037 | 0.036 | 0.039 | 0.027 | |

r | 0.93 | 0.92 | 0.93 | 0.93 | 0.92 | 0.95 | |

MBE | −0.003 | −0.003 | −0.004 | −0.003 | −0.006 | −0.003 | |

VG-PDI | RMSE | 0.048 | 0.052 | 0.056 | 0.048 | 0.051 | 0.038 |

MAE | 0.038 | 0.041 | 0.047 | 0.038 | 0.04 | 0.026 | |

r | 0.92 | 0.91 | 0.89 | 0.92 | 0.92 | 0.96 | |

MBE | −0.008 | −0.008 | −0.007 | −0.008 | −0.008 | −0.006 | |

VG_{m}-PDI | RMSE | 0.067 | 0.064 | 0.056 | 0.058 | 0.061 | 0.048 |

MAE | 0.049 | 0.049 | 0.043 | 0.045 | 0.048 | 0.033 | |

r | 0.84 | 0.87 | 0.89 | 0.89 | 0.88 | 0.92 | |

MBE | 0.011 | 0.006 | 0.002 | 0.011 | 0.006 | 0.006 |

_{m}: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively; RMSE: root mean square error (cm

^{3}cm

^{−3}), MAE: mean absolute error (cm

^{3}cm

^{−3}), r: correlation coefficient, MBE: mean bias error (cm

^{3}cm

^{−3}).

**Table 5.**Performance evaluation of the bimodal and bimodal-PDI parametric PTFs for all modeling scenarios (S1–S6).

Models | S1 | S2 | S3 | S4 | S5 | S6 | |
---|---|---|---|---|---|---|---|

K-b | RMSE | 0.067 | 0.071 | 0.074 | 0.066 | 0.072 | 0.063 |

MAE | 0.052 | 0.055 | 0.060 | 0.052 | 0.057 | 0.045 | |

r | 0.88 | 0.87 | 0.84 | 0.88 | 0.86 | 0.90 | |

MBE | 0.030 | 0.030 | 0.028 | 0.029 | 0.030 | 0.026 | |

VG-b | RMSE | 0.083 | 0.085 | 0.085 | 0.085 | 0.089 | 0.085 |

MAE | 0.065 | 0.067 | 0.067 | 0.067 | 0.070 | 0.064 | |

r | 0.88 | 0.87 | 0.86 | 0.86 | 0.85 | 0.88 | |

MBE | 0.016 | 0.017 | 0.010 | 0.015 | 0.016 | 0.014 | |

VG_{m}-b | RMSE | 0.098 | 0.099 | 0.085 | 0.100 | 0.092 | 0.086 |

MAE | 0.076 | 0.077 | 0.065 | 0.079 | 0.070 | 0.064 | |

r | 0.81 | 0.80 | 0.85 | 0.80 | 0.82 | 0.88 | |

MBE | 0.056 | 0.059 | 0.047 | 0.059 | 0.048 | 0.059 | |

FX-b-PDI | RMSE | 0.059 | 0.059 | 0.066 | 0.06 | 0.06 | 0.047 |

MAE | 0.046 | 0.047 | 0.052 | 0.048 | 0.046 | 0.033 | |

r | 0.90 | 0.90 | 0.87 | 0.90 | 0.91 | 0.94 | |

MBE | 0.002 | 0.004 | 0.002 | 0.007 | 0.003 | 0.004 | |

K-b-PDI | RMSE | 0.058 | 0.069 | 0.065 | 0.061 | 0.071 | 0.056 |

MAE | 0.046 | 0.053 | 0.053 | 0.048 | 0.055 | 0.04 | |

r | 0.89 | 0.86 | 0.86 | 0.89 | 0.85 | 0.92 | |

MBE | 0.012 | 0.013 | 0.012 | 0.011 | 0.012 | 0.014 | |

VG-b-PDI | RMSE | 0.063 | 0.076 | 0.074 | 0.064 | 0.081 | 0.066 |

MAE | 0.050 | 0.058 | 0.060 | 0.050 | 0.063 | 0.046 | |

r | 0.91 | 0.88 | 0.86 | 0.91 | 0.87 | 0.92 | |

MBE | −0.007 | −0.005 | −0.009 | −0.008 | −0.007 | −0.007 | |

VG_{m}-b-PDI | RMSE | 0.065 | 0.069 | 0.071 | 0.065 | 0.071 | 0.059 |

MAE | 0.053 | 0.055 | 0.058 | 0.052 | 0.057 | 0.044 | |

r | 0.89 | 0.89 | 0.86 | 0.90 | 0.88 | 0.93 | |

MBE | 0.031 | 0.030 | 0.029 | 0.030 | 0.031 | 0.030 |

_{m}: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively; RMSE: root mean square error (cm

^{3}cm

^{−3}), MAE: mean absolute error (cm

^{3}cm

^{−3}), r: correlation coefficient, MBE: mean bias error (cm

^{3}cm

^{−3}).

Models | par | S1 | S2 | S3 | S4 | S5 | S6 |
---|---|---|---|---|---|---|---|

BC | α | 0.73 | 0.73 | 0.74 | 0.73 | 0.63 | 0.73 |

λ | 0.15 | 0.28 | 0.10 | 0.15 | 0.36 | 0.59 | |

θ_{r} | 0.57 | 0.76 | 0.57 | 0.57 | 0.76 | 0.82 | |

θ_{s} | 0.73 | 0.81 | 0.61 | 0.73 | 0.41 | ||

FX | α | 0.70 | 0.64 | 0.63 | 0.70 | 0.64 | 0.49 |

n | 0.49 | 0.45 | 0.49 | 0.28 | 0.55 | 0.55 | |

h_{r} | −0.18 | −0.15 | −0.18 | −0.09 | −0.15 | −0.09 | |

θ_{s} | 0.72 | 0.79 | 0.61 | 0.72 | 0.82 | ||

m | 0.65 | 0.65 | 0.66 | 0.66 | 0.53 | 0.77 | |

FX-PDI | α | 0.71 | 0.71 | 0.72 | 0.71 | 0.71 | 0.62 |

n | 0.73 | 0.73 | 0.73 | 0.73 | 0.73 | 0.74 | |

θ_{r} | 0.79 | 0.84 | 0.79 | 0.77 | 0.84 | 0.89 | |

θ_{s} | 0.72 | 0.80 | 0.62 | 0.72 | 0.83 | ||

m | 0.35 | 0.17 | 0.30 | 0.33 | 0.25 | 0.34 | |

FX-b-PDI | α_{1} | 0.14 | 0.14 | 0.14 | 0.12 | −0.04 | 0.06 |

n_{1} | −0.06 | −0.12 | 0.11 | −0.06 | −0.12 | −0.08 | |

θ_{r} | 0.62 | 0.80 | 0.64 | 0.74 | 0.80 | 0.82 | |

θ_{s} | 0.73 | 0.80 | 0.68 | 0.73 | 0.84 | ||

α_{2} | 0.46 | 0.47 | 0.51 | 0.19 | 0.43 | 0.31 | |

n_{2} | −0.14 | −0.19 | 0.06 | −0.21 | −0.09 | −0.32 | |

w_{2} | 0.04 | −0.28 | 0.20 | −0.18 | −0.11 | −0.12 | |

m_{1} | 0.36 | 0.21 | 0.42 | 0.36 | 0.21 | 0.21 | |

m_{2} | −0.34 | −0.34 | 0.09 | 0.20 | −0.21 | 0.05 | |

K | h_{m} | 0.84 | 0.85 | 0.84 | 0.84 | 0.85 | 0.81 |

σ | 0.27 | 0.27 | 0.18 | 0.27 | 0.17 | 0.33 | |

θ_{r} | 0.77 | 0.79 | 0.78 | 0.77 | 0.81 | 0.90 | |

θ_{s} | 0.73 | 0.81 | 0.61 | 0.73 | 0.83 | ||

K-PDI | h_{m} | 0.83 | 0.86 | 0.83 | 0.83 | 0.83 | 0.86 |

σ | 0.53 | 0.51 | 0.57 | 0.53 | 0.42 | 0.47 | |

θ_{r} | 0.87 | 0.89 | 0.88 | 0.87 | 0.89 | 0.95 | |

θ_{s} | 0.72 | 0.80 | 0.77 | 0.72 | 0.78 | ||

K-b | h_{m1} | −0.04 | −0.04 | −0.04 | −0.27 | −0.27 | −0.10 |

σ_{1} | −0.23 | −0.23 | −0.23 | −0.23 | −0.23 | −0.21 | |

θ_{r} | 0.03 | −0.03 | 0.11 | 0.03 | 0.06 | 0.31 | |

θ_{s} | 0.72 | 0.80 | 0.61 | 0.72 | 0.82 | ||

h_{m2} | 0.08 | 0.17 | 0.13 | 0.24 | 0.29 | 0.26 | |

σ_{2} | 0.10 | 0.13 | 0.13 | 0.10 | 0.22 | 0.23 | |

w_{2} | −0.39 | −0.39 | −0.39 | −0.39 | −0.17 | −0.26 | |

K-b-PDI | h_{m1} | −0.08 | −0.08 | −0.08 | −0.08 | −0.23 | −0.24 |

σ_{1} | −0.18 | −0.18 | −0.18 | −0.18 | −0.18 | −0.13 | |

θ_{r} | 0.80 | 0.80 | 0.80 | 0.80 | 0.80 | 0.88 | |

θ_{s} | 0.73 | 0.80 | 0.62 | 0.73 | 0.83 | ||

h_{m2} | 0.08 | 0.08 | 0.01 | 0.06 | −0.04 | 0.07 | |

σ_{2} | −0.34 | −0.34 | −0.34 | −0.06 | −0.06 | −0.03 | |

w_{2} | −0.40 | −0.49 | 0.08 | −0.34 | −0.24 | 0.02 | |

VG | α | 0.67 | 0.67 | 0.68 | 0.67 | 0.59 | 0.64 |

n | 0.13 | 0.18 | 0.10 | 0.10 | 0.14 | 0.40 | |

θ_{r} | 0.74 | 0.80 | 0.69 | 0.73 | 0.79 | 0.78 | |

θ_{s} | 0.73 | 0.81 | 0.60 | 0.73 | 0.82 | ||

VG-PDI | α | 0.65 | 0.65 | 0.66 | 0.65 | 0.64 | 0.50 |

n | 0.55 | 0.55 | 0.56 | 0.55 | 0.56 | 0.55 | |

θ_{r} | 0.89 | 0.89 | 0.88 | 0.89 | 0.89 | 0.94 | |

θ_{s} | 0.72 | 0.80 | 0.62 | 0.72 | 0.83 | ||

VG-b | α_{1} | −0.36 | −0.36 | 0.08 | −0.36 | −0.01 | −0.04 |

n_{1} | 0.05 | 0.05 | 0.04 | 0.05 | 0.04 | 0.07 | |

θ_{r} | 0.10 | 0.30 | 0.06 | 0.10 | 0.30 | 0.22 | |

θ_{s} | 0.73 | 0.80 | 0.61 | 0.73 | 0.83 | ||

α_{2} | 0.16 | 0.08 | 0.16 | −0.14 | −0.25 | 0.14 | |

n_{2} | −0.02 | −0.02 | −0.02 | −0.02 | −0.02 | −0.02 | |

w_{2} | −0.21 | −0.21 | −0.21 | −0.21 | −0.21 | −0.21 | |

VG-b-PDI | α_{1} | −0.07 | −0.26 | −0.27 | −0.07 | −0.28 | −0.17 |

n_{1} | −0.01 | −0.09 | −0.03 | −0.29 | −0.32 | −0.32 | |

θ_{r} | 0.41 | 0.39 | 0.41 | 0.41 | 0.38 | 0.40 | |

θ_{s} | 0.73 | 0.81 | 0.62 | 0.73 | 0.84 | ||

α_{2} | −0.03 | −0.05 | −0.03 | −0.02 | −0.08 | 0.08 | |

n_{2} | −0.06 | 0.19 | −0.06 | 0.06 | 0.20 | 0.05 | |

w_{2} | 0.10 | 0.03 | 0.10 | 0.10 | 0.03 | 0.03 | |

VG_{m} | α | 0.71 | 0.71 | 0.71 | 0.71 | 0.70 | 0.70 |

n | 0.60 | 0.60 | 0.60 | 0.60 | 0.60 | 0.58 | |

θ_{r} | 0.44 | 0.65 | 0.44 | 0.41 | 0.66 | 0.63 | |

θ_{s} | 0.72 | 0.79 | 0.59 | 0.72 | 0.82 | ||

m | 0.73 | 0.75 | 0.71 | 0.73 | 0.74 | 0.74 | |

VG_{m}-PDI | α | 0.64 | 0.64 | 0.69 | 0.64 | 0.69 | 0.69 |

n | 0.54 | 0.55 | 0.49 | 0.26 | 0.27 | 0.42 | |

θ_{r} | 0.83 | 0.83 | 0.83 | 0.83 | 0.83 | 0.88 | |

θ_{s} | 0.70 | 0.79 | 0.69 | 0.70 | 0.82 | ||

m | 0.27 | 0.34 | 0.51 | 0.34 | 0.43 | 0.42 | |

VG_{m}-b | α_{1} | −0.08 | −0.08 | −0.08 | 0.05 | 0.05 | −0.04 |

n_{1} | 0.20 | 0.21 | 0.07 | 0.20 | 0.11 | 0.17 | |

θ_{r} | 0.31 | 0.32 | 0.31 | 0.31 | 0.42 | 0.38 | |

θs | 0.69 | 0.78 | 0.67 | 0.69 | 0.81 | ||

α_{2} | 0.02 | −0.16 | 0.04 | −0.09 | 0.34 | 0.15 | |

n_{2} | −0.33 | −0.22 | −0.29 | −0.19 | −0.22 | −0.18 | |

w_{2} | −0.24 | −0.24 | −0.20 | −0.24 | −0.17 | −0.15 | |

m_{1} | −0.18 | 0.07 | 0.03 | 0.20 | 0.07 | 0.20 | |

m_{2} | 0.04 | 0.05 | 0.04 | −0.08 | 0.05 | 0.05 | |

VG_{m}-b-PDI | α_{1} | −0.16 | −0.16 | −0.16 | −0.16 | −0.16 | 0.37 |

n_{1} | 0.17 | 0.22 | 0.26 | 0.26 | 0.26 | 0.26 | |

θ_{r} | 0.27 | 0.25 | 0.27 | 0.27 | 0.25 | 0.31 | |

θ_{s} | 0.73 | 0.81 | 0.62 | 0.73 | 0.84 | ||

α_{2} | −0.33 | −0.35 | −0.31 | −0.31 | −0.36 | −0.28 | |

n_{2} | −0.27 | 0.13 | −0.27 | −0.27 | 0.13 | 0.27 | |

w_{2} | 0.14 | −0.04 | 0.08 | 0.08 | −0.04 | −0.21 | |

m_{1} | −0.19 | 0.31 | 0.19 | 0.21 | 0.12 | 0.14 | |

m_{2} | 0.11 | 0.16 | 0.11 | 0.00 | −0.02 | −0.04 |

_{m}: van Genuchten [24] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [30,31] and bimodal variants of the models, respectively; S1 to S6 denote modeling scenarios 1 to 6, respectively.

© 2020 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**

Haghverdi, A.; Öztürk, H.S.; Durner, W. Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: II. Evaluation of Parametric Pedotransfer Functions Against Direct Fits. *Water* **2020**, *12*, 896.
https://doi.org/10.3390/w12030896

**AMA Style**

Haghverdi A, Öztürk HS, Durner W. Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: II. Evaluation of Parametric Pedotransfer Functions Against Direct Fits. *Water*. 2020; 12(3):896.
https://doi.org/10.3390/w12030896

**Chicago/Turabian Style**

Haghverdi, Amir, Hasan Sabri Öztürk, and Wolfgang Durner. 2020. "Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: II. Evaluation of Parametric Pedotransfer Functions Against Direct Fits" *Water* 12, no. 3: 896.
https://doi.org/10.3390/w12030896