# Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: I. Direct Model Fit Using the Extended Evaporation and Dewpoint Methods

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{m}model). In addition, eleven alternative expressions including Peters–Durner–Iden (PDI), bimodal, and bimodal-PDI variants of the original models were evaluated. We used a data set consisting of 94 soil samples from Turkey and the United States with high-resolution measured data (a total of 9264 measured water retention data pairs) mainly via the HYPROP system and supplemented for some samples with measured dry-end data using the WP4C instrument. Among unimodal expressions, the FX and the K models with the Mean Absolute Error (MAE) values equal to 0.005 cm

^{3}cm

^{−3}and 0.015 cm

^{3}cm

^{−3}have the highest and the lowest accuracy, respectively. Overall, the alternative variants provided a better fit than the unimodal expressions. The unimodal models, except for the FX model, fail to provide reliable dry-end estimations using HYPROP data (average MAE: 0.041 cm

^{3}cm

^{−3}, average r: 0.52). Our results suggested that only models that account for the zero water content at the oven dryness and properly shift from the middle range to dry-end (i.e., the FX model and PDI variants) can adequately represent the full SWRC using typical data obtained via the HYPROP system.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Extended Evaporation Approach Using the HYPROP System

#### 2.2. Dew Point Technique Using WP4C System

#### 2.3. Data Collection and Laboratory Analyses

^{3}stainless-steel cylinders (inside diameter: 8 cm, height: 5 cm) were used to collect undisturbed samples. All samples were brought up to saturation and then two vertically aligned tensiometers were installed in two holes made via a small auger, such that the tensiometers’ tips were positioned 1.25 cm below and above the center of the cylinder (2.5 cm). For the Turkish samples, the air-dried disturbed sieved samples (10 mesh sieve; <2.0 mm) were packed and brought up to saturation under vacuum in the same size cylinders to dry BDs close to the field condition (repacking was done in multiple steps to ensure consistent BD of the repacked samples). Only the upper part of the samples was open to the atmosphere for evaporation. For each sample, depending on the soil type, the measurement campaign lasted between 4 and 14 days (on average, ten days per sample for the entire data set). Throughout the measurement campaign, the soil tensions were automatically monitored using HYPROP-VIEW software at the two depths using the tensiometers. The integral of the water content distribution over the entire column divided by the height of the sample was considered by HYPROP as the mean water content of the sample. The measurement frequencies of the tensiometers were set to be higher at the beginning of the experiment (approximately once a minute for the first hour) and further apart afterward (every ten minutes). The weights of the United States samples were continuously recorded, while the Turkish samples were weighed only twice a day following the single balance approach as outlined in the HYPROP operation manual [50].

^{3}) were sliced off the original undisturbed sample and stored in caped small sample cups (15 cm

^{3}capacity). The subsamples were taken from the top, middle and bottom of the original sample with different water contents. The soil tension of each subsample was measured using WP4C and the weight of the sample was immediately recorded afterward. Finally, the subsamples and the remaining soil material from HYPROP were put in the oven to get the oven-dry weight of the sample. The water retention data measured by the HYPROP and WP4C are shown in Figure 3. The HYPROP measurement campaign yielded high-resolution data (on average, 98 data points per soil sample) between saturation and up close to the wilting point with an average pF of 1.76. On average WP4C measurements had a pF value of 4.68.

#### 2.4. Soil Hydraulic Models

#### 2.4.1. Original Unimodal Expressions

_{m}model; [15]).

_{r}and θ

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

_{r}is the tension corresponding to θ

_{r}and h

_{0}is the soil tension at zero water content (in this study was set to 10

^{6.8}cm which is the suction at oven dryness for 105 °C, [53]) and e is the Euler number.

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

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

#### 2.4.2. 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:

#### 2.4.3. 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).

#### 2.4.4. 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.

#### 2.5. Model Parametrization

#### 2.6. Evaluation Statistics

^{3}cm

^{−3}), $\overline{E}$ and $\overline{M}$ are the mean fitted and the mean measured water content values and n is the number of the measured water retention points for all samples considered in each analysis.

## 3. Results

#### 3.1. Overall Performance of the Models

^{3}cm

^{−3}and 0.020 cm

^{3}cm

^{−3}. The MBE values are close to zero indicating systematic bias for none of the models. Among unimodal expressions, the FX and the K models with the MAE values equal to 0.005 cm

^{3}cm

^{−3}and 0.015 cm

^{3}cm

^{−3}have the highest and the lowest accuracy, respectively. The PDI variants outperformed the unimodal expressions for VG, VG

_{m}and K models. This improvement was more pronounced for clay and clay loam samples compared to loam and sandy loam samples when performance statistics were separately calculated (data not shown here) for the textures with highest number of samples (i.e., clay, clay loam, loam, sandy loam). When MAE for each sample is separately calculated (data not shown here), the FX, FX-PDI and VG

_{m}-PDI model exhibit the best fit to the measured data for 47%, 23% and 22% of our soil samples, respectively. The K model followed by the BC model show the worst fit (highest MAE values) for 73% and 26% of the samples, respectively. The lower performance of the K and BC models were observed across clay, clay loam, loam, and sandy loam textures (data not shown here).

^{3}cm

^{−3}and 0.012 cm

^{3}cm

^{−3}, respectively. When MAE values for individual US soil samples are considered (data not shown here), the VG

_{m}-b-PDI and K model exhibit the best and worst fit for 47% and 93% of the samples, respectively.

#### 3.2. Extrapolation Capability beyond HYPROP Range

^{3}cm

^{−3}and 0.023 cm

^{3}cm

^{−3}and r values vary between 0.57 and 0.97. Without the dry-end data, the MAE values vary from 0.008 cm

^{3}cm

^{−3}to 0.078 cm

^{3}cm

^{−3}and the correlation coefficients range between 0.42 and 0.96. The MBE values range from −0.001 to 0.018 with WP4C data and from 0.001 to 0.078 without WP4C data included in the fitting process.

_{m}models), indicating a low performance mainly due to overestimation of the dry-end data. The FX model shows comparable performance with and without the dry-end data included in the fitting process with the MAE values equal to 0.009 cm

^{3}cm

^{−3}and 0.011 cm

^{3}cm

^{−3}, respectively. All variants of the FX model show similar performance. Without dry-end data, for the K, the VG and the VG

_{m}models, the MAE values of the PDI, the bimodal, and the bimodal-PDI variants were substantially lower than that of the unimodal expressions. The performance improvement of these alternative variants over the original models is less pronounced with the dry-end data included in the fitting process. Except for the FX model, the none-PDI variants of the other models (i.e., the original expression as well as the bimodal variants) have an average r value of 0.56 showing a lower performance than the PDI variants with an average r value of 0.92.

#### 3.3. Performance Across Tension Classes

## 4. Discussion

#### 4.1. Unimodal Expressions

_{m}, and K models and equations developed by Tani [56], Russo [57], Rossi and Nimmo [58], Kosugi [12,13], and Assouline et al. [6]. They found that the VG

_{m}model with a mean RMSE value of 0.008 cm

^{3}cm

^{−3}showed the highest match with the measured water retention data providing the best fit for 67% of the soil samples. The BC, the VG and the K models in their study showed lower performance with mean RMSE values of 0.014 cm

^{3}cm

^{−3}, 0.010 cm

^{3}cm

^{−3}and 0.009 cm

^{3}cm

^{−3}, respectively. Among unimodal expressions in our study, the VG

_{m}model (RMSE: 0.011 cm

^{3}cm

^{−3}, MAE: 0.008 cm

^{3}cm

^{−3}) ranked as the second-best model after the FX model (RMSE: 0.007 cm

^{3}cm

^{−3}, MAE: 0.005 cm

^{3}cm

^{−3}). The K (RMSE: 0.020 cm

^{3}cm

^{−3}, MAE: 0.015 cm

^{3}cm

^{−3}) and the BC (RMSE: 0.016 cm

^{3}cm

^{−3}, MAE: 0.011 cm

^{3}cm

^{−3}) models showed the lowest overall performance in our study and the lowest performance for most of the soil textures (data not shown).

^{3}cm

^{−3}for pF < 2) is attributed to its discontinuous form, which causes a sharp corner where its two parts meet near saturation. The poor match of the BC model near-saturation was also reported by Cornelis et al. [33]. Our results also showed relatively high wet-end MAE values for the K model (MAE: 0.016 cm

^{3}cm

^{−3}for pF < 2), the VG model (MAE: 0.012 cm

^{3}cm

^{−3}for pF < 2) and less pronounced for the VG

_{m}model (MAE: 0.008 cm

^{3}cm

^{−3}for pF < 2), a trend that was not reported by Cornelis et al. [33]. The differences between the results of the two studies might be related to the limited number of measured data near saturation in the data set used by Cornelis et al. [33] as opposed to the high-resolution wet-end data that was utilized in this study.

^{3}cm

^{−3}, average r: 0.52), a drawback also highlighted by Cornelis et al. [33], hence their extrapolation beyond the driest measured water retention point is not suggested (Table 5). Our results show that when additional dry-end measurements are included in the fitting process, the unimodal models perform better in the SWRC dry-end (average dry-end MAE improved from 0.041 cm

^{3}cm

^{−3}to 0.014 cm

^{3}cm

^{−3}). We tested the feasibility of setting the residual water content to zero in the model data fitting scheme as a possible strategy to improve the dry-end estimation by the unimodal VG and VG

_{m}models. This strategy was ineffective, however, and caused a decrease in the overall performance of both models. For instance, the MAE values increased from 0.007 to 0.009 cm

^{3}cm

^{−3}for the VG model and from 0.005 to 0.006 cm

^{3}cm

^{−3}for the VG

_{m}model when the residual water content was set to zero. The FX model was the only unimodal expression that accounted for the zero water content at the oven dryness and accurately transitioned from the middle range to dry-end. A similar result was reported by Lu et al. [36] who studied the extrapolative capability of the FX model to estimate the complete SWRC for soils with various textures. They reported that when measured data pairs in the tension range of 0 to 1500 kPa are available, the fitted FX model provided a good agreement with experimental data from the complete SWRC. They, therefore, concluded that the need for the dry-end measurement can be eliminated with the FX model.

#### 4.2. Bimodal and PDI Variants

_{m}-b models (Table 5) reveal that despite their high number of free parameters, bimodal variants fail to accurately estimate the dry-end data using HYPROP data. The PDI variants, on the other hand, outperformed unimodal and bimodal variants in estimating the SWRC dry-end with average dry-end MAE values of 0.008 cm

^{3}cm

^{−3}(WP4C included in the fitting process) and 0.014 cm

^{3}cm

^{−3}(WP4C not included in the fitting process). Lu et al. [17] evaluated the performance of three models (developed by Fayer and Simmons, [59]; Webb [60]; Khlosi et al., [61]) describing the complete SWRC using data points in the pressure plate measurement range (i.e., tension less than 1500 kPa) on eight soils. The results of their study illustrated that when measured data pairs are available in the 0 to 1500-kPa range, all three models could provide a reliable fit to the complete SWRC range. Our results suggested that typical HYPROP data (i.e., high-resolution data from saturation to 100 kPa soil tension range and an additional point close to the wilting point) are sufficient for an accurate representation of the full SWRC (from saturation to oven-dryness) when PDI variants are fitted to the experimental data. Since all of our undisturbed samples were coarse-textured and the number of soil samples was relatively limited, we recommend further investigations to determine whether the same result holds for other soil types.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Model ^{1} | Par | C ^{2} | CL | L | LS | SCL | SL | SiL |
---|---|---|---|---|---|---|---|---|

BC | α | 0.051 | 0.045 | 0.015 | 0.060 | 0.040 | 0.026 | 0.006 |

λ | 0.309 | 0.252 | 0.329 | 0.704 | 0.287 | 0.352 | 0.385 | |

θr | 0.117 | 0.053 | 0.001 | 0.069 | 0.016 | 0.009 | 0.000 | |

θs | 0.651 | 0.603 | 0.523 | 0.418 | 0.373 | 0.428 | 0.533 | |

FX | α | 0.030 | 0.024 | 0.006 | 0.042 | 0.024 | 0.014 | 0.002 |

n | 4.571 | 3.517 | 1.509 | 5.818 | 2.970 | 2.373 | 1.926 | |

hr | 3135 | 465 | 1228 | 5238 | 359 | 606 | 25045 | |

θs | 0.654 | 0.607 | 0.532 | 0.422 | 0.381 | 0.435 | 0.549 | |

m | 0.244 | 0.244 | 0.782 | 0.443 | 0.332 | 0.625 | 0.752 | |

FX-PDI | α | 0.032 | 0.029 | 0.008 | 0.039 | 0.031 | 0.015 | 0.003 |

n | 3.233 | 2.552 | 1.317 | 4.899 | 2.580 | 1.904 | 1.327 | |

θr | 0.301 | 0.189 | 0.012 | 0.100 | 0.060 | 0.054 | 0.036 | |

θs | 0.651 | 0.604 | 0.530 | 0.420 | 0.378 | 0.433 | 0.543 | |

m | 0.453 | 0.237 | 0.798 | 0.703 | 0.183 | 0.697 | 1.000 | |

FX-b-PDI | α_{1} | 0.033 | 0.021 | 0.027 | 0.023 | 0.024 | 0.032 | 0.005 |

n_{1} | 5.014 | 4.845 | 4.269 | 3.995 | 5.225 | 3.917 | 3.491 | |

θr | 0.285 | 0.156 | 0.049 | 0.105 | 0.082 | 0.046 | 0.000 | |

θs | 0.652 | 0.605 | 0.531 | 0.421 | 0.380 | 0.435 | 0.547 | |

α_{2} | 0.031 | 0.023 | 0.009 | 0.043 | 0.014 | 0.013 | 0.010 | |

n_{2} | 4.896 | 1.671 | 4.758 | 5.997 | 2.550 | 6.159 | 1.668 | |

w_{2} | 0.469 | 0.671 | 0.602 | 0.717 | 0.549 | 0.325 | 0.420 | |

m_{1} | 0.370 | 0.665 | 0.789 | 0.774 | 0.701 | 0.610 | 0.469 | |

m_{2} | 0.490 | 0.369 | 0.657 | 0.789 | 0.758 | 0.808 | 0.787 | |

K | hm | 141.8 | 200.7 | 508.7 | 47.5 | 236.8 | 293.8 | 958.8 |

σ | 1.795 | 1.918 | 1.660 | 1.022 | 1.804 | 1.647 | 1.727 | |

θr | 0.227 | 0.183 | 0.086 | 0.096 | 0.084 | 0.070 | 0.058 | |

θs | 0.667 | 0.616 | 0.536 | 0.424 | 0.391 | 0.439 | 0.548 | |

K-PDI | hm | 77 | 105 | 428 | 40 | 90 | 219 | 846 |

σ | 0.971 | 1.232 | 1.517 | 0.678 | 1.083 | 1.260 | 1.504 | |

θr | 0.376 | 0.326 | 0.140 | 0.150 | 0.189 | 0.137 | 0.107 | |

θs | 0.653 | 0.608 | 0.532 | 0.426 | 0.382 | 0.434 | 0.544 | |

K-b | hm_{1} | 2386 | 2789 | 2105 | 40 | 947 | 453 | 1055 |

σ_{1} | 1.723 | 1.643 | 1.354 | 0.617 | 1.904 | 1.403 | 1.783 | |

θr | 0.039 | 0.025 | 0.051 | 0.018 | 0.039 | 0.042 | 0.034 | |

θs | 0.655 | 0.607 | 0.532 | 0.428 | 0.383 | 0.436 | 0.549 | |

hm_{2} | 3720 | 3447 | 440 | 5178 | 641 | 1462 | 859 | |

σ_{2} | 1.941 | 1.819 | 1.291 | 1.876 | 1.190 | 1.351 | 0.690 | |

w_{2} | 0.522 | 0.525 | 0.486 | 0.387 | 0.419 | 0.493 | 0.371 | |

K-b-PDI | hm_{1} | 371 | 1053 | 833 | 58 | 553 | 264 | 586 |

σ_{1} | 1.290 | 1.144 | 1.298 | 0.652 | 1.264 | 1.352 | 1.265 | |

θr | 0.346 | 0.248 | 0.144 | 0.143 | 0.153 | 0.121 | 0.145 | |

θs | 0.653 | 0.606 | 0.531 | 0.421 | 0.383 | 0.437 | 0.547 | |

hm_{2} | 300 | 629 | 269 | 41 | 104 | 392 | 695 | |

σ_{2} | 1.362 | 1.738 | 1.040 | 0.566 | 1.457 | 1.298 | 0.868 | |

w_{2} | 0.519 | 0.501 | 0.478 | 0.696 | 0.385 | 0.527 | 0.439 | |

VG | α | 0.030 | 0.026 | 0.008 | 0.047 | 0.023 | 0.015 | 0.004 |

n | 1.519 | 1.397 | 1.459 | 2.292 | 1.469 | 1.515 | 1.486 | |

θr | 0.182 | 0.124 | 0.025 | 0.084 | 0.051 | 0.035 | 0.009 | |

θs | 0.660 | 0.611 | 0.531 | 0.448 | 0.387 | 0.436 | 0.543 | |

VG-PDI | α | 0.025 | 0.022 | 0.008 | 0.033 | 0.023 | 0.014 | 0.003 |

n | 2.152 | 1.813 | 1.492 | 3.174 | 1.872 | 1.776 | 1.561 | |

θr | 0.377 | 0.322 | 0.062 | 0.150 | 0.175 | 0.100 | 0.048 | |

θs | 0.653 | 0.607 | 0.530 | 0.425 | 0.382 | 0.433 | 0.542 | |

VG-b | α_{1} | 0.012 | 0.010 | 0.012 | 0.026 | 0.013 | 0.010 | 0.135 |

n_{1} | 2.118 | 1.806 | 3.164 | 2.038 | 3.461 | 2.939 | 2.157 | |

θr | 0.010 | 0.000 | 0.040 | 0.007 | 0.004 | 0.018 | 0.039 | |

θs | 0.653 | 0.607 | 0.532 | 0.433 | 0.380 | 0.434 | 0.552 | |

α_{2} | 0.014 | 0.018 | 0.013 | 0.125 | 0.016 | 0.026 | 0.005 | |

n_{2} | 2.364 | 2.112 | 5.743 | 3.642 | 2.534 | 3.095 | 2.056 | |

w_{2} | 0.512 | 0.515 | 0.486 | 0.471 | 0.471 | 0.440 | 0.557 | |

VG-b-PDI | α_{1} | 0.016 | 0.015 | 0.023 | 0.036 | 0.015 | 0.025 | 0.014 |

n_{1} | 2.619 | 2.158 | 4.393 | 3.181 | 4.170 | 4.183 | 1.642 | |

θr | 0.218 | 0.070 | 0.107 | 0.137 | 0.097 | 0.079 | 0.158 | |

θs | 0.652 | 0.605 | 0.531 | 0.420 | 0.382 | 0.435 | 0.547 | |

α_{2} | 0.020 | 0.014 | 0.009 | 0.023 | 0.025 | 0.042 | 0.006 | |

n_{2} | 1.954 | 2.005 | 3.127 | 4.892 | 2.078 | 2.518 | 2.842 | |

w_{2} | 0.547 | 0.472 | 0.609 | 0.541 | 0.552 | 0.613 | 0.514 | |

VG_{m} | α | 0.048 | 0.041 | 0.004 | 0.053 | 0.038 | 0.021 | 0.001 |

n | 4.549 | 3.300 | 1.278 | 8.287 | 4.354 | 2.814 | 1.036 | |

θr | 0.136 | 0.086 | 0.039 | 0.073 | 0.024 | 0.028 | 0.058 | |

θs | 0.653 | 0.625 | 0.524 | 0.445 | 0.377 | 0.433 | 0.549 | |

m | 0.095 | 0.089 | 0.596 | 0.246 | 0.086 | 0.309 | 0.917 | |

VG_{m}-PDI | α | 0.036 | 0.033 | 0.007 | 0.036 | 0.030 | 0.016 | 0.001 |

n | 4.596 | 3.480 | 1.379 | 5.768 | 3.019 | 2.563 | 1.139 | |

θr | 0.371 | 0.296 | 0.083 | 0.145 | 0.135 | 0.095 | 0.102 | |

θs | 0.651 | 0.604 | 0.530 | 0.422 | 0.379 | 0.433 | 0.546 | |

m | 0.248 | 0.187 | 0.655 | 0.421 | 0.316 | 0.450 | 0.994 | |

VG_{m}-b | α_{1} | 0.009 | 0.004 | 0.017 | 0.031 | 0.014 | 0.008 | 0.004 |

n_{1} | 3.705 | 2.504 | 7.352 | 1.985 | 4.525 | 4.815 | 8.349 | |

θr | 0.009 | 0.009 | 0.050 | 0.000 | 0.011 | 0.027 | 0.002 | |

θs | 0.651 | 0.605 | 0.532 | 0.441 | 0.380 | 0.435 | 0.548 | |

α_{2} | 0.021 | 0.019 | 0.010 | 0.266 | 0.028 | 0.010 | 0.008 | |

n_{2} | 2.896 | 2.378 | 2.869 | 6.018 | 4.969 | 4.374 | 4.742 | |

w_{2} | 0.537 | 0.523 | 0.586 | 0.487 | 0.400 | 0.554 | 0.517 | |

m_{1} | 0.712 | 0.756 | 0.536 | 0.568 | 0.467 | 0.571 | 0.330 | |

m_{2} | 0.485 | 0.535 | 0.590 | 0.167 | 0.447 | 0.600 | 0.446 | |

VG_{m}-b-PDI | α_{1} | 0.014 | 0.018 | 0.015 | 0.023 | 0.016 | 0.012 | 0.005 |

n_{1} | 2.403 | 2.701 | 6.140 | 1.725 | 1.256 | 2.969 | 8.223 | |

θr | 0.195 | 0.131 | 0.112 | 0.129 | 0.128 | 0.090 | 0.051 | |

θs | 0.652 | 0.605 | 0.531 | 0.422 | 0.382 | 0.436 | 0.546 | |

α_{2} | 0.019 | 0.021 | 0.059 | 0.043 | 0.020 | 0.063 | 0.006 | |

n_{2} | 3.252 | 3.705 | 3.788 | 9.167 | 7.448 | 8.085 | 6.824 | |

w_{2} | 0.504 | 0.445 | 0.507 | 0.381 | 0.321 | 0.393 | 0.516 | |

m_{1} | 0.389 | 0.236 | 0.639 | 0.671 | 0.583 | 0.596 | 0.400 | |

m_{2} | 0.591 | 0.556 | 0.500 | 0.567 | 0.713 | 0.459 | 0.334 |

^{1}BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG: Van Genuchten [15] constrained and VG

_{m}: Van Genuchten [15] unconstrained unimodal soil water retention models. PDI and b denote Peters–Durner–Iden [10,14] and bimodal variants of the models, respectively;

^{2}C: Clay, CL: Clay loam, SCL: Sandy clay loam, L: Loam, LS: Loamy sand, SL: Sandy loam, Sil: Silt loam.

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**Figure 1.**Hydraulic Property Analyzer (HYPROP) system components (

**right**) and the four typical phases (P1 to P4) of a measurement campaign (

**left**) for the optimal measuring curve including the regular measurement range phase (P1), the boiling delay phase (P2), the cavitation phase (P3) and the air entry phase (P4), adapted from HYPROP user manual [50].

**Figure 2.**WP4C components: Views of the front and inside of the soil chamber block adapted from WP4C Dew Point PotentioMeter Operator’s Manual [51].

**Figure 3.**Soil water retention data pairs measured by the HYPROP and WP4C (

**left**) and soil textural classes (

**right**) where green and red represent Turkish and USA samples, respectively.

**Figure 4.**Scatter plots of fitted versus measured soil water content for all 16 models. The while and blue circles depict Turkish and USA data, respectively. BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG and VG

_{m}: van Genuchten [15] constrained and unconstrained unimodal soil water retention models. PDI and B denote Peters–Durner–Iden [10,14] and bimodal variants of the models, respectively.

**Figure 5.**Performance of the 16 models at different suctions. BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG and VG

_{m}: van Genuchten [15] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [10,14] and bimodal variants, respectively. pF classes (C) tensions (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).

Attributes ^{1} | Min. | Max. | SD ^{2} | Average |
---|---|---|---|---|

Sand (%) | 5.90 | 83.60 | 19.23 | 39.57 |

Silt (%) | 4.43 | 57.60 | 9.75 | 29.06 |

Clay (%) | 0.01 | 62.20 | 15.70 | 31.37 |

BD (Mg m^{−3}) | 0.91 | 1.82 | 0.24 | 1.31 |

pF | −2.00 | 6.20 | 0.83 | 1.78 |

θ (cm^{3} cm^{−3}) | 0.01 | 0.69 | 0.14 | 0.44 |

θs (cm^{3} cm^{−3}) | 0.31 | 0.69 | 0.11 | 0.56 |

^{1}BD: soil bulk density; θ: soil water content measured by the HYPROP and WP4C instrument; θs: saturated water content; pF: Log (h) where h is the soil tension (cm water column);

^{2}SD: standard deviation.

Models ^{1} | Parameters | Min | Max | Unit |
---|---|---|---|---|

FX-b-PDI, K-b, K-b-PDI, VG-b, VG-b-PDI, VG_{m}-b, VG_{m}-b-PDI | w_{2} | 0 | 1 | - |

BC, FX, FX-PDI, FX-b-PDI, K, K-PDI, K-b, K-b-PDI, VG, VG-PDI, VG-b, VG-b-PDI, VG_{m}, VG_{m}-PDI, VG_{m}-b, VG_{m}-b-PDI | θ_{s} | 0.1 | 1 | cm^{3} cm^{−3} |

BC, FX-PDI, FX-b-PDI, K, K-PDI, K-b, K-b-PDI, VG, VG-PDI, VG-b, VG-b-PDI, VG_{m}, VG_{m}-PDI, VG_{m}-b, VG_{m}-b-PDI | θ_{r} | 0 | 0.4 | cm^{3} cm^{−3} |

K, K-PDI, K-b, K-b-PDI | σ, σ_{1}, σ_{2} | 0.2 | 6 | - |

BC, FX, FX-PDI, FX-b-PDI, VG, VG-PDI, VG-b, VG-b-PDI, VG_{m}, VG_{m}-PDI, VG_{m}-b, VG_{m}-b-PDI | α, α_{1}, α_{2} | 0.00001 | 0.5 | cm^{−1} |

K, K-PDI | h_{m} | 1 | 10,000 | - |

K-b, K-b-PDI | h_{m}_{1}, h_{m}_{2} | 1 | 100,000 | cm^{−1} |

FX | h_{r} | 1 | 100,000 | cm |

BC | λ | 0.1 | 10 | - |

FX-PDI, FX-b-PDI, VG_{m}, VG_{m}-PDI, VG_{m}-b, VG_{m}-b-PDI | m, m_{1}, m_{2} | 0.01 | 1 | - |

FX | m | 0.1 | 10 | - |

FX | n | 0.1 | 10 | - |

FX-PDI, FX-b-PDI, VG, VG-PDI, VG-b, VG-b-PDI, VG_{m}-b, VG_{m}-b-PDI | n, n_{2} | 1.01 | 15 | - |

VG_{m}, VG_{m}-PDI, VG_{m}-b, VG_{m}-b-PDI | n, n_{1} | 0.5 | 15 | - |

Model ^{1} | MBE ^{2} | MAE | RMSE | r |
---|---|---|---|---|

BC | 0.001 | 0.011 | 0.016 | 0.994 |

FX | 0.000 | 0.005 | 0.007 | 0.999 |

FX-PDI | 0.000 | 0.005 | 0.007 | 0.999 |

K | 0.004 | 0.015 | 0.020 | 0.991 |

K-PDI | 0.001 | 0.008 | 0.011 | 0.997 |

VG | 0.003 | 0.011 | 0.015 | 0.995 |

VG-PDI | 0.000 | 0.007 | 0.009 | 0.998 |

VG_{m} | 0.002 | 0.008 | 0.011 | 0.997 |

VG_{m}-PDI | 0.000 | 0.005 | 0.007 | 0.999 |

^{1}BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG and VG

_{m}: van Genuchten [15] constrained and unconstrained unimodal soil water retention models. PDI denotes Peters–Durner–Iden [10,14] variants of the models;

^{2}MBE: mean bias error (cm

^{3}cm

^{−3}), RMSE: root mean square error (cm

^{3}cm

^{−3}), MAE: mean absolute error (cm

^{3}cm

^{−3}), r: correlation coefficient.

Model ^{1} | MBE ^{2} | MAE | RMSE | r |
---|---|---|---|---|

BC | 0.000 | 0.006 | 0.008 | 0.995 |

FX | 0.000 | 0.004 | 0.005 | 0.998 |

FX-PDI | 0.000 | 0.004 | 0.005 | 0.998 |

FX-b-PDI | 0.000 | 0.002 | 0.004 | 0.999 |

K | 0.000 | 0.009 | 0.012 | 0.989 |

K-PDI | −0.001 | 0.006 | 0.008 | 0.996 |

K-b | 0.000 | 0.003 | 0.005 | 0.998 |

K-b-PDI | 0.000 | 0.003 | 0.004 | 0.999 |

VG | 0.000 | 0.007 | 0.009 | 0.993 |

VG-PDI | 0.000 | 0.005 | 0.006 | 0.997 |

VG-b | 0.000 | 0.003 | 0.005 | 0.999 |

VG-b-PDI | 0.000 | 0.002 | 0.004 | 0.999 |

VG_{m} | 0.000 | 0.005 | 0.007 | 0.996 |

VG_{m}-PDI | 0.000 | 0.004 | 0.005 | 0.998 |

VG_{m}-b | 0.000 | 0.003 | 0.004 | 0.999 |

VG_{m}-b-PDI | 0.000 | 0.002 | 0.004 | 0.999 |

^{1}BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG and VG

_{m}: van Genuchten [15] constrained and unconstrained unimodal soil water retention models. PDI and b denote Peters–Durner–Iden [10,14] and bimodal variants of the models, respectively;

^{2}MBE: mean bias error (cm

^{3}cm

^{−3}), RMSE: root mean square error (cm

^{3}cm

^{−3}), MAE: mean absolute error (cm

^{3}cm

^{−3}), r: correlation coefficient.

**Table 5.**Performance of the 16 models at the Soil Water Retention Curve (SWRC) dry-end (pF range: 3.86–6.20, pF average: 4.68) with and without WP4C measurements included in the fitting process.

Model ^{1} | WP4C Data Included | WP4C Data Excluded | ||||||
---|---|---|---|---|---|---|---|---|

r | MBE | MAE | RMSE ^{2} | r | MBE | MAE | RMSE | |

BC | 0.907 | 0.007 | 0.011 | 0.014 | 0.694 | 0.018 | 0.021 | 0.028 |

FX | 0.950 | 0.005 | 0.009 | 0.011 | 0.923 | 0.008 | 0.011 | 0.015 |

FX-PDI | 0.956 | 0.005 | 0.008 | 0.010 | 0.920 | 0.009 | 0.013 | 0.016 |

FX-b-PDI | 0.975 | 0.001 | 0.005 | 0.007 | 0.956 | 0.006 | 0.010 | 0.012 |

K | 0.572 | 0.018 | 0.023 | 0.030 | 0.422 | 0.078 | 0.078 | 0.082 |

K-PDI | 0.934 | 0.017 | 0.018 | 0.020 | 0.905 | 0.022 | 0.023 | 0.026 |

K-b | 0.955 | −0.001 | 0.006 | 0.009 | 0.534 | 0.005 | 0.023 | 0.030 |

K-b-PDI | 0.964 | 0.004 | 0.007 | 0.009 | 0.919 | 0.011 | 0.013 | 0.017 |

VG | 0.820 | 0.012 | 0.015 | 0.021 | 0.454 | 0.063 | 0.063 | 0.068 |

VG-PDI | 0.957 | 0.010 | 0.011 | 0.013 | 0.910 | 0.017 | 0.018 | 0.022 |

VG-b | 0.974 | 0.002 | 0.005 | 0.007 | 0.584 | 0.010 | 0.024 | 0.033 |

VG-b-PDI | 0.974 | 0.002 | 0.005 | 0.007 | 0.946 | 0.004 | 0.009 | 0.011 |

VG_{m} | 0.922 | 0.006 | 0.010 | 0.014 | 0.509 | 0.031 | 0.032 | 0.041 |

VG_{m}-PDI | 0.960 | 0.005 | 0.007 | 0.010 | 0.899 | 0.012 | 0.015 | 0.018 |

VG_{m}-b | 0.972 | 0.000 | 0.005 | 0.007 | 0.725 | 0.001 | 0.016 | 0.023 |

VG_{m}-b-PDI | 0.975 | 0.001 | 0.005 | 0.007 | 0.950 | 0.002 | 0.008 | 0.010 |

^{1}BC: Brooks and Corey [7], FX: Fredlund and Xing [9], K: Kosugi [12], VG and VG

_{m}: van Genuchten [15] constrained and unconstrained unimodal models. PDI and b denote Peters–Durner–Iden [10,14] and bimodal variants of the models, respectively;

^{2}MBE: mean bias error (cm

^{3}cm

^{−3}), RMSE: root mean square error (cm

^{3}cm

^{−3}), MAE: mean absolute error (cm

^{3}cm

^{−3}), r: correlation coefficient.

© 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/).

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**MDPI and ACS Style**

Haghverdi, A.; Najarchi, M.; Öztürk, H.S.; Durner, W. Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: I. Direct Model Fit Using the Extended Evaporation and Dewpoint Methods. *Water* **2020**, *12*, 900.
https://doi.org/10.3390/w12030900

**AMA Style**

Haghverdi A, Najarchi M, Öztürk HS, Durner W. Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: I. Direct Model Fit Using the Extended Evaporation and Dewpoint Methods. *Water*. 2020; 12(3):900.
https://doi.org/10.3390/w12030900

**Chicago/Turabian Style**

Haghverdi, Amir, Mohsen Najarchi, Hasan Sabri Öztürk, and Wolfgang Durner. 2020. "Studying Unimodal, Bimodal, PDI and Bimodal-PDI Variants of Multiple Soil Water Retention Models: I. Direct Model Fit Using the Extended Evaporation and Dewpoint Methods" *Water* 12, no. 3: 900.
https://doi.org/10.3390/w12030900