# Developing Pseudo Continuous Pedotransfer Functions for International Soils Measured with the Evaporation Method and the HYPROP System: II. The Soil Hydraulic Conductivity Curve

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## Abstract

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_{NN}-PTF) approach and assessed its accuracy and reliability using two independent data sets with hydraulic conductivity measured via the evaporation method. The primary data set contained 150 international soils (6963 measured data pairs), and the second dataset consisted of 79 repacked Turkish soil samples (1340 measured data pairs). Four models with different combinations of the input attributes, including soil texture (sand, silt, clay), bulk density (BD), and organic matter content (SOM), were developed. The best performing international (root mean square error, RMSE = 0.520) and local (RMSE = 0.317) PTFs only had soil texture information as inputs when developed and tested using the same data set to estimate log(K). However, adding BD and SOM as input parameters increased the reliability of the international PC

_{NN}-PTFs when the Turkish data set was used as the test data set. We observed an overall improvement in the performance of PTFs with the increasing number of data points per soil textural class. The PC

_{NN}-PTFs consistently performed high across tension ranges when developed and tested using the international data set. Incorporating the Turkish data set into PTF development substantially improved the accuracy of the PTFs (on average close to 60% reduction in RMSE). Consequently, we recommend integrating local HYPROP

^{TM}(Hydraulic Property Analyzer, Meter Group Inc., USA) data sets into the international data set used in this study and retraining the PC

_{NN}-PTFs to enhance their performance for that specific region.

## 1. Introduction

_{s}) information. A popular four-parameter expression developed by van Genuchten [4] is widely used for SWRC parametrization, which coupled with Mualem-van Genuchten model [4,5] is often used for SHCC parametrization using the K

_{s}as a scaling factor. The SHCC can also be described by Gardner’s empirical expression [6], which in some cases works similarly or even better than the Mualem-van Genuchten model [6].

_{s}(point PTF) and parameters of the van Genuchten water retention model (parametric PTF), which are subsequently used for estimating the SHCC using the abovementioned approach [7,8,9,10]. For example, Schaap and Leij [11] used PTF-based SWRC parameters of van Genuchten equation to estimate the SHCC. They observed that the best results are obtained when the parameters K

_{s}and L (a term for the interaction between pore size and tortuosity) were flexible and not fixed as is the case in the classical Mualem–van Genuchten model. PTFs estimating unsaturated hydraulic conductivity at specific moisture tensions also exist (e.g., Moosavi and Sepaskhah [12]). However, little is known about the development and application of PTFs to directly estimate the SHCC [13].

_{s}are used to estimate the SHCC. The first type of error is associated with estimating the parameters of the van Genuchten model and K

_{s}using parametric and point PTFs, respectively. The second type of error is related to the Mualem–van Genuchten parameterization of the SWRC and SHCC, which is often fitted only using a few water retention data pairs measured by equilibrium approaches. The SHCC estimations via Mualem-van Genuchten model can result in poor performance near saturation because of the inability to account for water flow through macropores [10,14,15]. Furthermore, K

_{s}is a highly variable soil hydraulic property dependent upon the pore geometry at the scale of interest [14] and seasonal variability [15]. Significant variabilities in K

_{s}estimations might occur when using different PTFs modeling approaches [16] and measurement techniques [17], ultimately reflected in the SHCC estimations.

^{TM}(Hydraulic Property Analyzer, Meter Group Inc., USA) system. The HYPROP system (Figure 1) is an automated evaporation-based benchtop laboratory system that works based on the extended evaporation method [21,22]. The HYPROP has a relatively fast measurement cycle and provides high resolution reliable simultaneous measurements of soil water content and unsaturated hydraulic conductivity within a few days or weeks [23,24,25]. Haghverdi et al. [26] used an HYPROP measured Turkish soil data set to develop water retention PC

_{NN}-PTFs and reported promising results. In a companion paper (Singh et al. [27]), we utilized the Schindler and Müller [20] dataset to develop water retention PC

_{NN}-PTFs. However, no PC

_{NN}-PTF has been developed to estimate the SHCC using high-resolution data measured via the evaporation method. Consequently, this study was carried out to (I) develop PC

_{NN}-PTFs for SHCC estimations by utilizing the abovementioned international [20] and Turkish [26] data sets measured via the evaporation method, (II) determine the accuracy and reliability of the PC

_{NN}-PTFs, and (III) assess the performance of the developed models across soil textures and different ranges of soil tension.

## 2. Materials and Methods

#### 2.1. Soil Data Sets

_{NN}-PTFs and evaluate their accuracy and reliability. The measured hydraulic conductivity data and the soil textural classification of the samples for both data sets are shown in Figure 2. The primary data set, hereafter referred to as the international data set, was published by Schindler & Müller [20] and consisted of 173 soils collected from 71 sites from all over the world. This data set contains the measurements of water retention, unsaturated hydraulic conductivity, K(h), and several basic soil properties, including textural data, organic matter content (SOM), and dry bulk density (BD). The soil hydraulic properties were measured using the evaporation experiments or the extended evaporation method via the HYPROP method. A majority of the soil samples in the data set were collected from arable lands, yet few samples from other land use types such as urban land, grassland, forests, fallow lands and riverbanks were also present. After screening the international data set, a subset of samples (i.e., 150 soils with 6963 total K(h) data pairs) was selected to develop PC

_{NN}-PTFs. The characteristics of the selected soils are shown in Table 1. The most dominant texture was silt loam; comprising 78 soil samples (52% of the data set), followed by loam; consisting of 18 soil samples (12% of the data set). Values of K(h) were log-transformed because hydraulic conductivity data are generally log-normally distributed [11]. The measured log(K(h)) values ranged from −6.64 to 0.98 (0 to 9.65 cm d

^{−1}), with an average of −2.26 (0.073 cm d

^{−1}). The pF (logarithmic transformation of soil tension in cm of water) values ranged from 0.22 to 4.21, with an average of 2.47.

_{s}data (pF 0) were measured using the falling head method with the KSAT instrument (Meter Group Inc., Pullman, WA, USA). The K(h) points were measured for each sample with pF ranging from 1.80 to 3.91, with an average of 2.51, and log(K) ranging from −4.75 to 3.27 (0 to 1862 cm d

^{−1}), with an average of −1.6 (0.03 cm d

^{−1}). Clay was the dominant texture (38 soil samples or 48.1% of the data set), followed by sandy loam (13 samples or 16.5% of the data set). Further details about the laboratory procedures used to develop this data set are available in Haghverdi et al. [21,28,29,30]. More information about HYPROP’s measurement principles is available in Schindler et al. [28].

#### 2.2. Unsaturated Hydraulic Conductivity Calculations

_{i}[cm/d]) between time points t

_{i}

_{-1}and t

_{i}through a horizontal plane that laid exactly in the middle of the two tension-tips:

_{i}is the change in water volume in the whole sample (cm

^{3}), Δt

_{i}is the time interval between two consecutive measurement points, and A the cross-sectional area (cm

^{2}) of the column.

_{i}(cm) is the time- and space-averaged tension, Δh

_{i}is the difference of tensions between the two tensiometer tips, and Δz (cm) is the distance between the tensiometer tips. The calculations assume that moisture tension and water content distribute linearly through the column and, therefore, the arithmetic mean of the tensions at two points was used. This simplified assumption was shown to provide accurate results because linearity errors in fluxes and tensions cancel each other out. [23]. The effect of hysteresis on water flow and transport is well understood [30]. However, since HYPROP measurements are taken during natural evaporation-based drying of soil samples, only drying hydraulic path was considered in this study.

#### 2.3. PC_{NN}-PTFs Development

_{NN}-PTF models and 20% for testing the models. The bootstrap technique was used on the development set to generate 100 replica datasets, each containing approximately 67% of the data. The rest of the development data (~33%) were used for cross-validation of the NN models. The training process was terminated when the root mean square error (RMSE) of the cross-validation subset began to increase or remain unchanged. To find the optimal topology of the neural network, the number of neurons of the hidden layer was iteratively changed from 1 to 14. This process was repeated five times leaving aside a different fold as the test set each time, such that all samples in the data set were used for testing the models. The outputs of the 100 PC

_{NN}-PTFs with optimum topology were averaged to obtain the hydraulic conductivity estimations.

#### 2.4. Modeling Scenarios

_{NN}-PTFs (developed using the international and the Turkish data sets) with four combination models of the input attributes, including textural constituents—sand, silt, and clay (SSC), BD, and SOM (Table 2). Model 1 constituted all the input attributes and the logarithmic transformation of soil suction (pF). Model 2 included SSC and pF. Model 3 included SSC, BD, and pF. Model 4 included SSC, SOM, and pF. The log(K (cm/d)) was the output parameter corresponding to the input pF value.

_{NN}-PTFs derived using the international data set. The results of the modeling scenarios were assessed to (i) quantify the improvements in international PC

_{NN}-PTFs for a specific region after incorporating local samples into the training data set and (ii) determining whether the international PC

_{NN}-PTFs trained using the integrated data works as accurately as the local PC

_{NN}-PTFs.

#### 2.5. Model Evaluation

_{NN}-PTFs:

## 3. Results

#### 3.1. Importance of the Input Predictors

_{NN}-PTFs developed in this study using different combinations of input predictors. All models showed acceptable performance, demonstrated by the well-scattered data around the 1:1 reference line except for the K

_{s}estimations in scenario 3 (training: the international dataset, test: Turkish datasets).

#### 3.2. Performance across Soil Textures

_{NN}-PTF models for the dominant soil textures, representing about 89% and 92% of the international and Turkish data sets, respectively. When the international data set was used as the training set (scenario 1), clay loam had higher RMSE and MAE values than other soil textures. RMSE values ranged from 0.517 to 1.124, MAE values ranged from 0.342 to 0.748, and MBE values ranged from 0.026 to 0.288 for all textures. Furthermore, the model showed a tendency to overestimate log(K) for all soil textures, except loam, where underestimation of log(K) was observed. The correlation coefficient (R) values varied between 0.603 in clay loam to 0.881 for silt loam.

#### 3.3. Performance at the Wet, Intermediate and Dry Parts of the SHCC

_{NN}-PTFs over three moisture ranges of the SHCC for the four data partitioning scenarios evaluated in this study. When the international data set was used for the training and testing of the models (scenario 1), the RMSE of Model 1 varied from 0.548 in the wet range to 0.603 in the dry range. The MAE values varied from 0.420 in the wet range to 0.440 in the intermediate range of the SHCC. The MBE values varied between −0.060 in the wet range to 0.140 in the dry range. The R values ranged from 0.509 for the wet to 0.640 in the intermediate range.

_{NN}-PTF models (scenario 2), the lowest error was observed in the intermediate range (RMSE = 0.317, MAE = 0.206, and MBE = 0.016) and the highest error belonged to the wet range (RMSE = 0.588, MAE = 0.471, and MBE = 0.031). The agreement between the observed and estimated log(K) was comparable among models with R ranging from a minimum of 0.809 in the wet range to a maximum of 0.860 in the intermediate range.

## 4. Discussion

#### 4.1. Accuracy and Reliability of the Developed PTFs

_{NN}-PTF showed the accuracy (same data set for development and test) and reliability (independent data sets for development and test) of RMSE = 0.520 and 1.097, respectively. The local PC

_{NN}-PTF developed and tested using the Turkish dataset showed even higher performance, as expected, with an RMSE of 0.317. Parasuraman et al. (2006) stated that better performance in estimating K

_{s}is observed when a NN model is trained even on a small set of relevant data rather than a larger general dataset. Our study emphasizes that a local data set, when available, should be included in the training of PC

_{NN}-PTF for a more accurate estimation of the SHCC.

_{NN}-PTF developed in this study. Børgesen et al. [7] reported a reasonable accuracy for their hydraulic conductivity PTFs with RMSE ranging from 0.598 to 1.196, yet most of the models showed underestimation. The above mentioned studies used the typical procedure to estimate the SHCC, which relies on estimated or measured K

_{s}values and parametric SWRC PTFs using NN approach combined with bootstrap. Therefore, the PC

_{NN}-PTFs approach developed and tested for the first time in this study could be used as an alternative high-performance approach to estimate the SHCC.

_{s}measurements because of factors such as sample size, soil conditions, flow geometry and installation procedures [32]. The same is true for the various devices used for unsaturated hydraulic conductivity measurements such as the steady-state pressure membrane method, tension disc infiltrometer, hot-air methods, and the widely used multistep outflow method [33,34,35]. We recommend using HYPROP data sets for developing hydraulic conductivity PC

_{NN}-PTFs. PC

_{NN}-PTF takes advantage of the high resolution measured data provided by the HYPROP system to learn the shape of the SHCC directly from the actual measured data points, unlike the parametric PTFs where the relationships between the parameters and their predictors have to be known a priori. Furthermore, using only one method (evaporation experiment) for obtaining hydraulic conductivity data in the laboratory is expected to improve the performance of the PC

_{NN}-PTFs by eliminating the variance related to employing multiple measurement techniques.

#### 4.2. Importance of Input Variables

_{s}is explained by properties like soil texture, porosity, SOM, and BD [36,37]. According to Zhang and Schaap [38], adding BD and SOM as input predictors improved the performance of PTFs in most studies estimating the K

_{s}. Moosavi and Sepaskhah [12] observed that the combination of inputs SSC, BD, SOM (Model 1 in this study) and SSC (Model 2 in this study) produced the best accuracy in estimating unsaturated hydraulic conductivity. In our study, considering SOM and BD as extra input attributes in addition to soil texture did not improve the accuracy of the international and local PTFs (scenarios 1 and 2). However, adding BD and SOM as input attributes noticeably enhanced the performance of the PC

_{NN}-PTF in scenarios 3 and 4. This result differs from our observation in the companion paper (Singh et al. [27]), where adding SOM as an extra input did not improve the performance of the water retention PC

_{NN}-PTFs.

_{s}is influenced primarily by porosity and macro water-stable aggregates, which are not among typical PTF inputs. Further studies are needed to determine the impact of considering additional soil structural input parameters on the performance of hydraulic conductivity PC

_{NN}-PTF.

#### 4.3. Performance across Textural Classes and Tension Ranges

_{NN}-PTFs showed better performance for fine-textured (Clay and Clay Loam) than more coarse-textured (Loam and Sandy Loam) Turkish soils (scenarios 2, 3, and 4), which is attributed to a relatively higher number of fine-textured Turkish soil samples. We observed an overall improvement in the performance of PTFs with the increasing number of data points per soil textural class (Figure 5). These results concur with the results we reported in the companion paper (Singh et al. [27]), where high performance was observed for the dominant soil textures for the SWRC estimations. Since PC

_{NN}-PTFs are machine learning-based models, their performance is expected to improve as more data become available for training.

_{NN}-PTFs was consistent across tension ranges when developed and tested using the international data set, which concurs with the results observed in the companion paper for soil water retention estimations [27]. Moosavi and Sepaskhah [12] reported a relatively lower accuracy of the NN-based PTFs to estimate hydraulic conductivity at the saturated and/or near-saturated tensions. We observed a somewhat higher error in the wet tension region (pF ≤ 2) for the Turkish data set, primarily when PC

_{NN}-PTFs were developed using the international data set. In our companion paper, however, the performance of water retention PC

_{NN}-PTFs was similar in three tension regions. The relatively higher error in the wet range in this study was because the Turkish data set only contained K

_{s}data in the wet part (measured via the KSAT instrument), while K

_{s}measurements were not available for the international data set.

## 5. Conclusions

_{NN}-PTFs to estimate the SHCC measured using the evaporation experiments, mainly via the HYPROP system. The PC

_{NN}-PTF approach showed promising performance for continuous hydraulic conductivity estimation over a wide range of soil tensions. The HYPROP system offers the advantage of producing high-resolution soil hydraulic conductivity data over a wide range of soil tensions (pF = 1.5 to 3.5), which is critical for developing robust PC

_{NN}-PTF models since this approach learns the shape of the SHCC directly from measured data. The KSAT instrument can be employed to measure the saturated hydraulic conductivity (K

_{s}) that can be used along with HYPROP data. The water retention PC

_{NN}-PTFs developed and validated in the first part of this study (Singh et al. [27]) also performed very well. Consequently, we recommend the PC

_{NN}-PTF approach to derive the next generation of water retention and hydraulic conductivity models using high-resolution data measured via the HYPROP system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The soil hydraulic conductivity and tension pairs (

**a**), and soil textural distribution for the datasets (

**b**). Dark orange circles depict the international data set [20] and blue circles represent the Turkish dataset [21,28,29,30]. pF is the logarithmic transformation of soil tension in cm of water and K is the unsaturated hydraulic conductivity.

**Figure 3.**Scatterplots of measured versus estimated log(K) using PC

_{NN}-PTFs. S1: training and test: the international dataset, S2: Training and test: Turkish dataset, S3: training: the international dataset, test: Turkish dataset, S4: training: international + Turkish dataset, test: Turkish dataset. Model 1 inputs: sand, silt and clay percentages (SSC), bulk density (BD), and soil organic matter content (SOM); Model 2 inputs: SSC; Model 3 inputs: SSC, BD; Model 4 inputs: SSC and SOM.

**Figure 4.**The root of mean squared error (RMSE) as a function of bulk density (BD) and organic matter content (SOM) for the PC

_{NN}PTF Model 1 with SSC, BD, SOM as inputs. The model was developed using combined international and Turkish data sets and tested using the Turkish data set (scenario 4). The error was calculated for each soil sample separately.

**Figure 5.**The root of mean squared error (RMSE) as a function of the number of measured hydraulic conductivity data pairs for each textural class for the PC

_{NN}-PTF Model 1 with SSC, BD, SOM as inputs. The model was developed using combined international and Turkish data sets.

**Table 1.**Characteristics of soils from international and Turkish data sets used in this study to develop and test pseudo continuous neural network pedotransfer functions (PC

_{NN}-PTFs).

International Data (150 Soil Samples with 6963 Data Pairs) | Turkish Data (79 Soil Samples with 1340 Data Pairs) | |||||
---|---|---|---|---|---|---|

Attribute | Mean | Range | SD | Mean | Range | SD |

Clay (%) | 20.0 | 0.0–60.0 | 12.5 | 34.1 | 9.4–62.2 | 15.1 |

Silt (%) | 56.4 | 0.2–86.8 | 17.1 | 30.7 | 5.2–57.6 | 8.7 |

Sand (%) | 23.6 | 3.9–99.8 | 17.4 | 35.3 | 6.0–84.0 | 17.4 |

Bulk density (g cm^{−3}) | 1.3 | 0.6–1.7 | 0.2 | 1.0 | 0.7–1.3 | 0.1 |

Organic matter content (%) | 3.1 | 0.0–12.0 | 2.5 | 1.2 | 0.0–3.1 | 0.6 |

Model | Input Attributes |
---|---|

1 | SSC, BD, SOM, pF |

2 | SSC, pF |

3 | SSC, BD, pF |

4 | SSC, SOM, pF |

^{3}cm

^{−3}), SOM: soil organic matter content (%), pF: the logarithmic transformation of soil tension in cm of water.

**Table 3.**Different data partitioning scenarios used in the study to train, test, and validate PC

_{NN}-PTFs.

Scenario | Data Sets |
---|---|

S1 | Training: International, Test: International. |

S2 | Training: Turkish, Test: Turkish. |

S3 | Training: International, Test: Turkish. |

S4 | Training: International + Turkish, Test: Turkish. |

**Table 4.**Performance of the PC

_{NN}-PTFs estimating log-transformed soil hydraulic conductivity data (cm d

^{−1}) across four modeling scenarios.

Training and Test: I | Training and Test: T | Training: I.; Test: T | Training: I + T; Test: T | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M | RMSE | MAE | MBE | R | RMSE | MAE | MBE | R | RMSE | MAE | MBE | R | RMSE | MAE | MBE | R |

1 | 0.571 | 0.428 | 0.013 | 0.855 | 0.343 | 0.227 | 0.023 | 0.959 | 1.097 | 0.971 | −0.959 | 0.935 | 0.429 | 0.312 | −0.139 | 0.947 |

2 | 0.520 | 0.406 | 0.027 | 0.881 | 0.317 | 0.217 | 0.011 | 0.965 | 1.317 | 1.254 | −1.249 | 0.954 | 0.613 | 0.456 | −0.335 | 0.906 |

3 | 0.547 | 0.418 | 0.033 | 0.868 | 0.336 | 0.219 | 0.017 | 0.961 | 1.235 | 1.142 | −1.133 | 0.942 | 0.453 | 0.308 | −0.165 | 0.938 |

4 | 0.529 | 0.417 | 0.022 | 0.877 | 0.350 | 0.243 | 0.043 | 0.958 | 1.243 | 1.144 | −1.132 | 0.943 | 0.554 | 0.400 | −0.280 | 0.920 |

**Table 5.**Performance of PC-PTFs on main textural classes of the international and Turkish data sets for estimating log(K).

Data Sets | Texture | RMSE | MAE | MBE | R |
---|---|---|---|---|---|

Training: I; Test: I | SiL | 0.517 | 0.421 | 0.026 | 0.881 |

L | 0.612 | 0.485 | −0.180 | 0.789 | |

SiCL | 0.433 | 0.342 | 0.144 | 0.870 | |

CL | 1.124 | 0.748 | 0.288 | 0.603 | |

SL | 0.593 | 0.522 | 0.042 | 0.700 | |

Training: T, Test: T | C | 0.252 | 0.174 | 0.006 | 0.978 |

SL | 0.366 | 0.277 | −0.096 | 0.966 | |

CL | 0.206 | 0.146 | 0.018 | 0.982 | |

L | 0.395 | 0.312 | −0.048 | 0.926 | |

Training: I, validation: T | C | 0.986 | 0.863 | −0.860 | 0.961 |

SL | 1.353 | 1.241 | −1.223 | 0.96 | |

CL | 0.964 | 0.894 | −0.894 | 0.978 | |

L | 1.444 | 1.377 | −1.370 | 0.917 | |

Training: I and T, Test: T | C | 0.303 | 0.218 | −0.013 | 0.972 |

SL | 0.535 | 0.446 | −0.317 | 0.963 | |

CL | 0.230 | 0.173 | −0.094 | 0.985 | |

L | 0.745 | 0.683 | −0.614 | 0.915 |

**Table 6.**Performance of the PC

_{NN}-PTFs (inputs: SSC, BD, OM, and pF) developed to estimate the log(K) at wet (pF ≤ 2) intermediate (2 < pF ≤ 3) and dry (pF > 3) parts of the SHCC.

Training and Test: I | Training and Test: T | Training: I, Validation: T | Training: I and T, Test: T | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Wet | Mid | Dry | Wet | Mid | Dry | Wet | Mid | Dry | Wet | Mid | Dry | |

RMSE | 0.548 | 0.570 | 0.603 | 0.588 | 0.317 | 0.375 | 2.285 | 1.002 | 0.466 | 0.757 | 0.400 | 0.396 |

MAE | 0.420 | 0.440 | 0.381 | 0.471 | 0.206 | 0.298 | 2.158 | 0.936 | 0.342 | 0.602 | 0.291 | 0.322 |

MBE | −0.060 | 0.007 | 0.140 | 0.031 | 0.016 | 0.109 | −2.158 | −0.926 | −0.292 | −0.520 | −0.134 | 0.154 |

R | 0.509 | 0.640 | 0.522 | 0.809 | 0.860 | 0.831 | 0.768 | 0.785 | 0.860 | 0.805 | 0.791 | 0.818 |

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## Share and Cite

**MDPI and ACS Style**

Singh, A.; Haghverdi, A.; Öztürk, H.S.; Durner, W.
Developing Pseudo Continuous Pedotransfer Functions for International Soils Measured with the Evaporation Method and the HYPROP System: II. The Soil Hydraulic Conductivity Curve. *Water* **2021**, *13*, 878.
https://doi.org/10.3390/w13060878

**AMA Style**

Singh A, Haghverdi A, Öztürk HS, Durner W.
Developing Pseudo Continuous Pedotransfer Functions for International Soils Measured with the Evaporation Method and the HYPROP System: II. The Soil Hydraulic Conductivity Curve. *Water*. 2021; 13(6):878.
https://doi.org/10.3390/w13060878

**Chicago/Turabian Style**

Singh, Amninder, Amir Haghverdi, Hasan Sabri Öztürk, and Wolfgang Durner.
2021. "Developing Pseudo Continuous Pedotransfer Functions for International Soils Measured with the Evaporation Method and the HYPROP System: II. The Soil Hydraulic Conductivity Curve" *Water* 13, no. 6: 878.
https://doi.org/10.3390/w13060878