Prediction of Soil Field Capacity and Permanent Wilting Point Using Accessible Parameters by Machine Learning

Abstract

The field capacity (FC) and permanent wilting point (PWP) are fundamental hydrological properties critical for assessing water availability within soils, rather than direct measures of soil health. Due to the challenges associated with their field measurement, alternative assessment methods are necessary. In this study, global-scale accessible soil data were retrieved from the world soil database called the World Soil Information Service (WoSIS), and artificial neural network (ANN) and gene-expression programming (GEP) algorithms were used to predict soil FC and PWP based on easily obtainable parameters from the database. The best-fit variable combination for FC (longitude, latitude, altitude, sand content, silt content, clay content, and electrical conductivity) and PWP (best-fit FC combination plus pH) modeling was determined. Both ANN and GEP showed greater accuracy than linear-based models in simulating the FC and PWP from the best-fit variables. The mean absolute error (MAE) was reduced by 51.54% for the FC and 56.38% for the PWP by the ANN model, compared with the linear model used in the previous literature. The normalized root mean square error (NRMSE) evaluation indicated that the ANN model performed best for PWP prediction (NRMSE of 19.9%), while the GEP model was superior for FC prediction (NRMSE of 29.9%). Between the ANN and GEP models, the ANN model showed a slightly higher model of interpretability; however, the GEP model exhibited a similar or better ability to avoid large error, based on the error distribution. Overall, our results demonstrated that machine learning is effective in predicting the FC and PWP from easily accessible data from WoSIS, and the GEP model is more preferable for FC and PWP modeling.

1. Introduction

Plant available water capacity provides information on the amount of water that is stored in soil and can be used by plants, which is determined by the differences between the soil field capacity (FC, −33 kPa of suction pressure) and the permanent wilting point (PWP, −1500 kPa of suction pressure) [1]. While the FC and PWP are critical for understanding soil water dynamics, it is important to clarify that these parameters primarily reflect the hydrological aspects of soil rather than direct measures of soil health. These properties are integral to applications such as irrigation management [2], crop modeling [3], and precision agriculture [4]. However, the FC and PWP of soil are often not available because their measurements are both time-consuming and costly [5,6]. In the latest soil health database, SoilHealthDB [7], only 81 results for the FC and PWP were reported out of 5908 data entries across 41 countries. This paucity of direct data underscores the urgent need for more efficient and cost-effective methodologies to estimate these crucial soil parameters. Therefore, the capability to estimate the FC and PWP from other easily measurable parameters is highly desirable [8]. Historically, pedotransfer functions (PTFs) have been applied to derive FC and PWP values based on parameters like soil texture, soil bulk density, and soil organic matter (SOM) [9,10]. These traditional methods, while noble in their intent, have been fraught with challenges, most prominently their limited accuracy. In an innovative push to enhance the precision of these models, parameters like the soil cation exchange capacity (CEC) [11], soil calcium carbonate (CaCO₃) content [12], geographical locations [13], and even geographical nuances [13] have been incorporated. Past efforts have shed light on the limitations of conventional models. For instance, Santra et al. [9] calculated the gravimetric FC and PWP in arid western India with sand, silt, and clay contents; organic carbon; and bulk density as input variables of PTF. The results showed that the coefficient of determination (R²) of best-fit FC and PWP was 0.63 and 0.34, respectively. Cueff et al. [14] used CEC and phosphorus content as additional inputs in addition to those used by Santra and colleagues to model the FC and PWP for soil in southwest France. The root mean squared error (RMSE) of their model ranged from 3.7% to 8.0% for the FC and 3.4% to 5.7% for the PWP. A significant drawback of previous efforts is the reliance on linear algorithms, which have inherently restricted the accuracy of these models [15,16].

Considering these challenges, there is a clear and present need for novel methodologies that move beyond traditional linear models, incorporating advanced algorithms that can capture the complexity and variability of soils, leading to more accurate and reliable estimations of the FC and PWP. For example, geostatistical approaches such as Kriging [17] and k-nearest neighbors (k-NN) [18] and parameter approaches such as regression tree (RT) [16], artificial neural network (ANN) [19], neuro-fuzzy (NF) [20], random forest (RF) [21], support vector machine (SVM) [22], gene-expression programming (GEP) [20], and deep learning (DL) [23] have been used for FC and PWP modeling in various studies. In a recent study, Shiri et al. [20] modeled FC and PWP for soil in Iran (0–60 cm) using five different algorithms: SVM, GEP, NF, RF, and RT. The contents of sand, silt, and clay; equivalent CaCO₃; bulk density; particle density; geometric mean of particle diameter; and geometric standard deviation of soil particle size were used as input variables. The study findings showed that the NF model had the highest R² followed by GEP. Taşan and Demir [12] simulated the FC and PWP in Turkey by conventional linear regression (REG) and an ANN, with sand, silt, and clay contents; bulk density; OM; CEC; and CaCO₃ as input variables. The results showed that the ANN model displayed higher modeling accuracy, and R², RMSE, and the mean absolute error (MAE) were 0.80, 3.12%, and 2.27% for the FC and 0.83, 1.84%, and 2.40% for the PWP, respectively. Yamaç et al. [23] modeled the FC and PWP of soil in Turkey using previously published PTFs, as well as k-NN, DL, and ANN using the same input data of sand, silt, and clay content; lime; bulk density; OM; particle density; and aggregate stability. The DL and ANN approaches had comparable R² and MAE values of 0.829 and 2.7% in the FC modeling. In PWP modeling, k-NN showed the best performance with an R² = 0.800 and MAE = 2.1%. Although previous studies demonstrated remarkable FC and PWP modeling ability by advanced algorithms, they were primarily based on site-specific datasets [14,24]. Therefore, models that can be applied to broader geographical regions are still missing.

Due to the importance of the FC and PWP in agricultural production, more reliable and broadly applicable FC and PWP modeling based on global-scale soil datasets is needed [12,13,14,23]. This study leveraged machine learning (ML) algorithms, specifically ANN and GEP, to predict FC and PWP values using an extensive range of data. The models incorporated physical (sand, silt, and clay content), chemical (electrical conductivity (EC) and pH), and geological (longitude, latitude, and altitude) parameters. Both EC and pH were found to be significantly related to soil water content (SWC) [25,26] and are easy to measure [27,28], but they were not used as modeling inputs in previous FC and PWP simulations.

ANNs were chosen for their proven ability to handle complex, non-linear interactions and large datasets, which makes them ideal for analyzing intricate relationships among various soil properties. Their effectiveness is supported by research that highlights their utility in modeling diverse soil dynamics, such as soil compaction and water retention [29,30]. Conversely, GEP was selected for its strengths in modeling non-linear and complex interactions, including soil water capacity [20] and soil stability [31], offering a robust framework for scenarios where traditional models may falter. This adaptability is crucial for fine-tuning models to uncover subtle patterns and interactions in the soil data. By integrating these advanced algorithms, this study aimed to enhance the predictive accuracy of FC and PWP models, providing a more comprehensive understanding of soil behavior that is critical for improving agricultural practices and irrigation strategies. The inclusion of ANNs and GEP represents a significant advancement over conventional linear models, offering new insights into soil dynamics that could lead to more effective and environmentally sustainable agricultural outcomes. Specific objectives of this study included the following:

(i)

Determine the optimal combination of variables for FC and PWP modeling;

(ii)

Apply machine learning algorithms to predict the FC and the PWP from easily measurable inputs from global scale accessible data.

2. Materials and Methods

2.1. Data Source and Process

Both the FC and PWP (cm³/100 cm³; %) and related soil composition (sand, silt, and clay contents; %), geographical location (longitude, latitude; decimal degree), pH, and EC (dS/m) were retrieved from the latest global soil database (WoSIS) [32]. Another geographical variable, altitude (m), was retrieved from Google Maps by the longitude and latitude information of the sample site. The WoSIS database includes over 5.8 million quality-assessed records from 173 countries. The procedure for data standardization and detailed information on the WoSIS database can be found in [32,33,34]. In this study, the WoSIS data were cleaned by only keeping information on the topsoil from 0 to 35 cm and by using a dataset that had complete information on the input variables, the FC, and the PWP. A total of 210 FC and 254 PWP values were extracted from the database for modeling. Descriptive statistical information for FC and PWP modeling, including the average (μ), standard deviation (σ), maximum (Max.) and minimum (Min.) of all used variables, and correlation coefficient (r) between each input variable and the FC or PWP, are summarized in

Table 1. Values of the input parameters for model fitting.

Target	Parameter	Input								Output
		Clay	Sand	Silt	Longitude	Latitude	Altitude	pH	EC	FC or PWP
		%	%	%	Decimal	Decimal	m	-	ds/m	%
FC (n = 210)	μ	24.314	53.401	21.405	23.519	−5.726	834.962	7.345	3.425	20.257
	σ	17.276	27.241	17.058	38.883	29.995	569.649	1.337	7.117	13.736
	Max.	79.000	98.000	78.000	116.721	69.433	2604.000	10.400	50.500	72.000
	Min.	2.000	1.000	0.000	−154.850	−33.821	−2.000	3.500	0.000	1.000
	r	0.618	−0.755	0.603	−0.180	0.623	−0.240	−0.093	−0.007	-
PWP (n = 254)	μ	24.348	53.361	21.536	15.843	−3.078	803.590	7.191	3.120	11.937
	σ	17.831	27.677	16.770	45.224	28.847	597.034	1.431	6.728	9.485
	Max.	79.000	98.000	78.000	116.721	69.433	2604.000	10.400	50.500	66.000
	Min.	2.000	1.000	0.000	−154.850	−33.821	−2.000	3.500	0.000	1.000
	r	0.706	−0.701	0.424	−0.187	0.410	−0.003	0.048	0.042	-

. The geographical distribution of the FC and PWP from the database are shown in Figure 1. The soil classification and countries from which the applied dataset for FC and PWP modeling were derived are shown in the Appendix A 

Table A1. FAO soil group and observed country in FC and PWP model.

Country	Target	Acrisols	Andosols	Arenosols	Calcisols	Cambisols	Chernozems	Ferralsols	Fluvisols	Kastanozems	Leptosols	Lixisols	Luvisols
Albania	FC												2
	PWP												2
Benin	FC
	PWP			1
Burkina Faso	FC											2
	PWP											2
Canada	FC					1							2
	PWP					1	2			1			2
Colombia	FC							5
	PWP							5
Ecuador	FC
	PWP								1
Ethiopia	FC
	PWP
Germany	FC					10	1		5				1
	PWP					10	1		5				1
India	FC					4
	PWP					4
Indonesia	FC	6
	PWP	6				1
Jamaica	FC
	PWP							1
Jordan	FC												1
	PWP												1
Kenya	FC							1					2
	PWP							1					2
Mozambique	FC	1		1				1	1
	PWP	1		1				1	1
Poland	FC
	PWP
Portugal	FC			2		2							1
	PWP			2		2							1
Puerto Rico	FC
	PWP	1
Sierra Leone	FC	1
	PWP	1
South Africa	FC	15		15	1	9		2	1	1	1	18	22
	PWP	15		15	1	9		2	1	1	1	18	22
Suriname	FC
	PWP	2						9
Sweden	FC
	PWP
Thailand	FC							1
	PWP							1
UK	FC
	PWP					1
Tanzania	FC		1			2
	PWP		1			2
USA	FC
	PWP
Zambia	FC	4
	PWP	4
Zimbabwe	FC												2
	PWP												2
Uncategorized	FC	4
	PWP	21

. The boxplot-based dataset distribution is illustrated in Figure A1.

2.2. Previous PTFs and Linear Regression Algorithm

Three regression-based PTFs applied in Saxton and Rawls [35], Adhikary et al. [36], and Tóth et al. [16] were collected as base models for FC and PWP modeling (Appendix B,

Table A3. Used parameters of each factor of published PTFs.

PTFs	Target	Soil	Silt	Clay	Organic Matters	Sa × OM	C × OM	Sa × C	Si × C	1/(OM + 1)	Si × OM’	C × OM’	Constant
		Sa	Si	C	OM					OM’
Saxton and Rawls	FC’	−0.251		0.195	0.011	0.006	−0.027	0.452					0.299
	FC	FC’ + (1.283(FC’)2 − 0.374(FC’) − 0.015)
Adhikary et al.	FC	−0.51	−0.27										56.37
Tóth et al.	FC		0.00154	0.00453					−0.000511	−0.1887	0.00144	0.00087	0.2449
Saxton and Rawls	PWP’	−0.024		0.487	0.006	0.005	−0.013	0.068					0.031
	PWP	PWP’ + (0.14(PWP’) − 0.02)
Adhikary et al.	PWP			0.44000									0.71
Tóth et al.	PWP		−0.00084	0.00213					0.000385	−0.0767	0.00095	0.00233	0.09878

). In addition, this study developed a WoSIS-based PTF model based on a linear regression algorithm, which is described as Equation (1).

$$FC \mathrm{o}\mathrm{r} PWP=\sum _{i=1}^n{\alpha }_i{x}_i+\beta +\epsilon$$

(1)

where n is the total number of data points, x_i is the ith input variable; α is the slope of the linear equation; β is an intercept of the regression, and ε is the error term.

2.3. Artificial Neural Networks (ANNs)

In addition to the four linear regression-based models, two ML-enabled algorithms, ANN and GEP, were also evaluated for their predictive accuracy of the FC and PWP. ANN is a powerful algorithm for non-linear problems that performs experience-based learning processes with an inherent punishment mechanism. Compared with a conventional linear model, an ANN deploys data features in multiple dimensions, which enables more comprehensive and accurate predictions. This study used NeuroSolution 7.1 software (NeuroDimension Inc, Gainesville, FL, USA) to develop predictive models for the FC and PWP by using a classic backpropagation neural network (BPNN). The maximum epoch was 1000 and the algorithm had a 0.01 learning rate. The Levenberg–Marquardt gradient search method with an early stopping callback was used to prevent overfitting. A single hidden layer was employed with 10 neuros [37]. The modeling dataset was normalized using Equation (2) and randomly divided into three sub-datasets for training, cross-validation (CV), and testing, with data ratios of 70%, 15%, and 15%, respectively. The training dataset was used for initial model development, and the CV dataset was used for hyper-parameter adjustment; the adjusted model was applied to the testing dataset for model performance evaluation [38]. Cross-validation was unavailable in the linear regression (REG) model; therefore, the dataset for CV was added to the training set for REG modeling. The data ratio of training and testing of the REG model was 85% and 15%, respectively.

$$x_{norm}={\displaystyle \frac{x-x_{min}}{{x_{max}-x}_{min}}}$$

(2)

where x_norm is the normalized dimensionless variable, x is the observed value, x_min is the minimum observed value, and x_max is the maximum observed value of the variable.

2.4. Gene-Expression Programming (GEP)

The GEP algorithm was the second ML-based algorithm investigated in this study. The algorithm is capable of establishing mathematical relationships between input variables and output parameters, similar to the gene algorithm (GA) and gene programming (GP). However, the computing is significantly faster and the computer accuracy is appreciably higher than the GA and GP [39]. A classic GEP begins with a major race and undergoes a continuous evolutionary process, such as selection, replication, mating, mutation, adaptation, reversal, and transformation to evolve toward a predetermined objective [40,41,42]. In this study, GeneXproTools 5.0 software (Gepsoft Ltd., Bristol, UK) was used for FC and PWP modeling. The algorithm was also trained, cross-validated, and tested by the same input dataset as the ANN model. The FC model incorporates two genes, whereas the PWP model is more intricate, encompassing five genes. The process of gene linkage, or how these genes interact and combine, is pivotal. For the FC model, a “minimum” linking function is employed, signifying that the smallest value or output from its genes is selected. Conversely, the PWP model utilizes a “multiplication” linking function, indicating that the outputs of its genes are multiplied together to produce a resultant value. Ten head sizes were used for both models with fifty chromosomes and ten thousand evolved generations with model elements including +, −, *,/, x⁻¹, x², x⁵, x^1/3, x^1/4, x^1/5, natural log (ln), floor, power, no processing, sine, cosine, tangent, arcsine, arctangent, average, and minimum. These operational elements, which can be likened to mathematical and computational functions, are integral to the model’s ability to adapt, learn, and ultimately solve the designated problem. Other parameters, such as the mutation rate, inversion rate, and gene transposition rate were set at default values, as described in the GEP theory in [43].

2.5. Assessment of the Best-Fit Combination of Input Variables for ML Based Models

The best-fit input variables for FC and PWP prediction using ML-based algorithms were determined using an ANN. Fifteen different combinations of input variables, including geographical location (L; longitude, latitude, and altitude), soil textures (S; sand, silt, and clay content), pH, EC, L + S, pH + EC, L + pH, L + EC, L + pH + EC, S + pH, S + EC, S + pH + EC, L + S + pH, L + S + EC, and L + S + pH + EC were evaluated. These combinations were used as input variables for ANN and typical model evaluation metrics, R², RMSE, normalized RMSE (NRMSE), and MAE [44,45,46], were calculated to determine the model performance of each combination, as shown in Equation (3)–(6). Furthermore, following the approach of Jamieson et al. [47], the NRMSE was categorized into four classes: simulations with NRMSE less than 10% were deemed excellent, those between 10% and 20% were good, those between 20% and 30% were fair, and those above 30% were considered poor.

$$R^2={\displaystyle \frac{{\sum }_{i=1}^n{\left({\widehat{y}}_i-\overline{y}\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}$$

(3)

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{n-1}}}$$

(4)

$$NRMSE={\displaystyle \frac{RMSE}{\overline{y}}}$$

(5)

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}AE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(6)

where R² is the coefficient of determination, ${\widehat{y}}_i$ is the predicted value in ith datum, $y_i$ is the observed value in ith datum, $\overline{y}$ is the average of the observations, n is the number of actual observations, RMSE is the root mean square error, NRMSE is normalized RMSE, MAE is the mean absolute error, and AE is the absolute error.

For the four linear models, Adhikary’s model [36] only used soil texture (sand, silt, and clay content) data as input variables, but the other two linear models required both OM and soil texture for modeling, which were also retrieved from the WoSIS database. A total of 210 data for the FC and 254 data for the PWP were used in the REG model and in Adhikary’s model. A total of 179 data in the FC and 219 data in the PWP were used in models in Saxton and Rawls [35] and Tóth et al. [16], the same as the dataset used in the REG, ANN, and GEP approaches in this study.

2.6. Rank the Input Variables for FC and PWP Modeling

The test dataset used for the FC and PWP simulation with ML models was evaluated by a Kolmogorov–Smirnov nonparametric test (K-S test) to determine the relative importance of input variables. The K-S test is a nonparametric test method for two sample comparison, which is unrelated to the sample’s frequency distribution and was used to compare the underlying probability distributions. The K-S test was analyzed by IBM SPSS software, version 22 (IBM Corp., Armonk, NY, USA). The test dataset was divided into 2 groups by MAE, with one group having an absolute error between observed targets and simulated results greater than MAE and the other group being smaller than MAE.

3. Results and Discussion

3.1. Determination of the Best-Fit Variables

To systematically elucidate the criteria behind the selection of variable combinations for FC and PWP modeling, our approach was grounded in a comprehensive analysis of soil properties’ interactions and their predictive significance. The modeling results from combinations of different inputs are shown in

Table 2. Performance of FC and PWP modeling by ANN using combinations of input variables.

Target	Combination	Training				CV				Testing
		R2	RMSE	NRMSE	MAE	R2	RMSE	NRMSE	MAE	R2	RMSE	NRMSE	MAE
FC	L	0.655	8.063	38.1%	6.338	0.707	9.908	44.8%	8.187	0.259	9.530	66.4%	8.019
	S	0.739	6.986	33.0%	4.979	0.722	9.164	41.5%	6.683	0.596	6.700	46.7%	5.178
	PH	0.016	13.787	65.2%	11.088	0.036	15.511	70.2%	11.964	0.013	11.666	81.3%	10.662
	EC	0.002	13.616	64.4%	11.146	0.013	15.406	69.7%	12.149	0.058	12.298	85.7%	11.254
	L + S	0.722	7.601	35.9%	5.115	0.777	9.008	40.8%	5.742	0.716	6.186	43.1%	4.578
	PH + EC	0.024	13.501	63.8%	10.931	0.054	15.748	71.3%	12.468	0.002	11.829	82.5%	10.891
	L + PH	0.747	6.925	32.7%	5.409	0.714	8.856	40.1%	7.070	0.279	11.271	78.6%	9.219
	L + EC	0.670	7.900	37.4%	6.254	0.613	10.220	46.3%	8.464	0.257	9.902	69.0%	8.328
	L + PH + EC	0.598	8.851	41.8%	7.280	0.707	10.223	46.3%	7.749	0.386	8.877	61.9%	7.457
	S + PH	0.775	6.511	30.8%	4.808	0.722	8.875	40.2%	6.888	0.575	7.062	49.2%	5.586
	S + EC	0.750	6.819	32.2%	5.165	0.696	8.823	39.9%	6.316	0.575	7.036	49.1%	4.574
	S + PH + EC	0.807	5.996	28.3%	4.489	0.696	8.928	40.4%	6.389	0.712	5.754	40.1%	4.137
	L + S + PH	0.727	7.403	35.0%	4.936	0.762	8.502	38.5%	5.186	0.752	5.705	39.8%	4.495
	L + S + EC	0.875	4.910	23.2%	3.683	0.763	8.013	36.3%	6.022	0.898	4.574	31.9%	3.133
	L + S + PH + EC	0.791	6.637	31.4%	4.598	0.698	9.554	43.2%	5.689	0.819	4.801	33.5%	3.647
PWP	L	0.151	9.154	74.6%	6.665	0.415	6.367	62.3%	4.887	0.398	7.781	64.4%	6.135
	S	0.514	6.877	56.0%	3.994	0.654	4.773	46.7%	3.742	0.790	4.327	35.8%	3.235
	PH	0.000	9.959	81.2%	7.194	0.015	8.055	78.8%	6.425	0.031	9.503	78.7%	7.787
	EC	0.006	9.957	81.1%	7.137	0.024	7.936	77.6%	6.304	0.070	9.845	81.5%	8.041
	L + S	0.639	5.939	48.4%	3.686	0.727	4.205	41.1%	3.195	0.834	3.840	31.8%	2.777
	PH + EC	0.007	9.904	80.7%	7.174	0.009	8.002	78.3%	6.411	0.087	9.855	81.6%	8.062
	L + PH	0.419	7.651	62.4%	5.649	0.473	5.962	58.3%	4.330	0.425	7.372	61.0%	6.125
	L + EC	0.537	6.941	56.6%	5.006	0.503	5.924	58.0%	4.458	0.382	7.719	63.9%	6.023
	L + PH + EC	0.487	7.248	59.1%	5.325	0.557	5.657	55.3%	4.386	0.370	7.857	65.1%	6.457
	S + PH	0.723	5.208	42.4%	3.520	0.697	4.502	44.0%	3.398	0.777	4.578	37.9%	3.384
	S + EC	0.535	6.721	54.8%	3.872	0.672	4.632	45.3%	3.575	0.785	4.400	36.4%	3.148
	S + PH + EC	0.596	6.310	51.4%	3.594	0.723	4.247	41.5%	3.049	0.813	4.188	34.7%	2.949
	L + S + PH	0.655	5.882	47.9%	3.555	0.784	3.894	38.1%	3.059	0.872	3.405	28.2%	2.508
	L + S + EC	0.662	4.910	40.0%	3.502	0.798	4.230	41.4%	2.890	0.877	3.578	29.6%	2.366
	L + S + PH + EC	0.660	6.031	49.1%	3.514	0.808	3.774	36.9%	2.778	0.915	2.442	19.9%	1.758

L: longitude, latitude, and altitude; S: sand, silt, and clay content; the bold font is the optimum combination in the field capacity and the permanent wilting point, respectively.

. The selection process was rigorously informed by a review of the existing literature and empirical evidence demonstrating the individual and combined effects of these properties on soil hydrology. In the FC model, the combination of L + S + EC resulted in the best performance. In the PWP simulation, L + S + pH + EC led to the highest accuracy. While the individual L and S datasets appeared to generate reasonable FC and PWP predictions, the combination of L + S generated more accurate FC and PWP simulations. Interestingly, while the model performance using ionic strength or EC alone as the input variable was quite low, their combination with L + S significantly enhanced the model performance in predicting both the FC and PWP. Previous studies suggested that EC is more sensitive to soil water variation [48,49] because the relative dielectric permittivity of water is generally more than an order of magnitude larger than that of other soil components [50]. As a result, the bulk relative dielectric permittivity of soil is primarily a function of the soil water content (SWC) [51]. The pH appears to have lower impact on the FC than the PWP, because the SWC easily fluctuates with the salinity content in the environment [52]. Chronically high or low pH has been found to cause soil water storage variation [26], but the SWC does not significantly change with rapidly varying pH. In addition, pH in the environment is relatively stable due to the carbonate buffering capacity, thus, its impact on SWC is not significant [53].

3.2. Comparison of Simulated FC and PWP by Different Models

GEP and REG models were developed based on the best-fit combination identified above. As a comparison, the same set of data used for these models was also applied to three PTFs from the literature mentioned earlier (i.e., Saxton and Rawls, Adhikary et al., and Tóth et al.). The outputs from the published PTFs, REG, ANN, and GEP models, are shown in Figure 2 and Figure 3 and Appendix C,

Table A4. Model performance of published PTFs, ANN, GEP, and REG.

Target	Model	Data			R2			RMSE			NRMSE			MAE
		Training	CV	Testing	Training	CV	Testing	Training	CV	Testing	Training	CV	Testing	Training	CV	Testing
FC	Saxton and Rawls	Total: 179			Total: 0.467			Total: 11.254			Total: 55.6%			Total: 8.120
	Adhikary et al.	Total: 210			Total: 0.527			Total: 9.969			Total: 49.2%			Total: 7.369
	Tóth et al.	Total: 179			Total: 0.347			Total: 17.225			Total: 85.0%			Total: 15.575
	REG	178	-	32	0.667	-	0.577	8.200	-	7.053	38.5%	-	49.2%	5.438	-	5.005
	ANN	146	32	32	0.875	0.763	0.898	4.910	8.013	4.574	23.2%	36.3%	31.9%	3.683	6.022	3.133
	GEP	146	32	32	0.700	0.753	0.843	7.487	8.337	4.290	35.4%	37.7%	29.9%	4.628	4.672	3.115
PWP	Saxton and Rawls	Total: 221			Total: 0.485			Total: 18.918			Total: 157.5%			Total: 14.392
	Adhikary et al.	Total: 254			Total: 0.501			Total: 17.699			Total: 147.3%			Total: 13.494
	Tóth et al.	Total: 221			Total: 0.472			Total: 16.244			Total: 135.2%			Total: 12.705
	REG	217	-	37	0.612	-	0.837	5.836	-	8.475	49.8%	-	69.0%	3.531	-	6.122
	ANN	180	37	37	0.660	0.808	0.915	6.031	3.774	2.442	49.1%	36.9%	19.9%	3.514	2.778	1.758
	GEP	180	37	37	0.852	0.665	0.889	3.723	4.551	2.746	30.8%	46.5%	22.3%	2.542	3.083	2.000

, respectively. The results showed that the model used in Adhikary et al. [36] has the highest R² in the published PTF in both the FC (0.683) and the PWP (0.823) simulation. In the ANN, GEP, and REG models, the ANN performed the best, as indicated by the R², RMSE, NRMSE, and MAE in both the FC and PWP models, followed by the GEP and REG model. ML-based ANN and GEP models exhibit a greater modeling accuracy than the REG and other three PTF models. The result is consistent with [20]. The RMSE, NRMSE, and MAE in the ML-based ANN model were decreased by 38.87%, 20.28%, and 51.54% in the FC simulation and 68.07%, 42.35%, and 56.38% in the PWP simulation compared with the PTF model used in Adhikary et al. The modeling results of each model can be found in

Table 3. Model input and performance parameters on the testing dataset.

Target	Model	Input Variables	Testing Dataset	R2	RMSE	NRMSE	MAE
FC	Saxton and Rawls	Sa, C, OM	28	0.644	8.077	56.3%	6.467
	Adhikary et al.	Sa, Si	32	0.683	7.482	52.2%	6.465
	Tóth et al.	Si, C, OM	28	0.490	17.766	123.9%	16.405
	REG	L, Sa, Si, C, EC	32	0.577	7.053	49.2%	5.005
	ANN	L, Sa, Si, C, EC	32	0.898	4.574	31.9%	3.133
	GEP	L, Sa, Si, C, EC	32	0.843	4.290	29.9%	3.115
PWP	Saxton and Rawls	Sa, C, OM	31	0.788	5.107	41.6%	3.567
	Adhikary et al.	C	37	0.823	7.646	62.2%	4.031
	Tóth et al.	Si, C, OM	31	0.776	4.632	37.7%	4.144
	REG	L, Sa, Si, C, EC, pH	37	0.837	8.475	69.0%	6.122
	ANN	L, Sa, Si, C, EC, pH	37	0.915	2.442	19.9%	1.758
	GEP	L, Sa, Si, C, EC, pH	37	0.889	2.746	22.3%	2.000

L: longitude, latitude, and altitude; Sa: sand; Si: silt; C: clay; OM: organic matters; EC: electrical conductivity.

and Appendix C, Figure A2. The expression equations derived from the GEP model for the FC and PWP models with the Python program are provided in Appendix D, and the equations of the FC and PWP developed with the REG are shown in Equations (7) and (8), respectively.

The simulation results for the different models in predicting the FC and PWP are summarized with their respective NRMSE values. For FC prediction, the models exhibited varying degrees of accuracy. According to the classification criteria adapted from Jamieson et al. [47], the GEP model, with an NRMSE of 29.9%, performed the best among all models and was classified as fair. The ANN model, with an NRMSE of 31.9%, was slightly higher but still falls into the poor category. Other models such as the REG, the model by Adhikary et al., and the model by Saxton and Rawls had NRMSE values of 49.2%, 52.2%, and 56.3%, respectively, and were all categorized as poor. Tóth et al. had the highest NRMSE at 123.9%, highlighting that the existing FC models may produce significant errors when using global scale datasets. For PWP prediction, the ANN model outperformed other models significantly, with an NRMSE of 19.9%, classifying it as good. The GEP model followed with an NRMSE of 22.3%, which is considered fair. Tóth et al. showed an NRMSE of 37.7%, falling into the poor category. The Saxton and Rawls, Adhikary et al., and REG models had NRMSE values of 41.6%, 62.2%, and 69.0%, respectively, and were all classified as poor. The NRMSE evaluation on the FC and PWP indicates that the ML-based algorithms provide a relatively accurate prediction.

$${\mathrm{F}\mathrm{C}}_{REG}=-0.3067·Lo+0.1836·La+0.0008·Al+0.3495·Sa+0.4724·Si+0.7345·C-0.093·EC-24.8313$$

(7)

$${PWP}_{REG}=-0.0474·Lo+0.0752·La+0.0004·Al+0.2925·Sa+0.3302·Si+0.6530·C-0.0049·EC+0.7750·pH-31.5174$$

(8)

where FC_REG is the field capacity and PWP_REG is the permanent wilting point in the REG model (%); Lo is the longitude; La is the latitude; Al is the altitude (m); Sa is the sand content (%); Si is the silt content (%); C is the clay content (%); and EC is the electrical conductivity (ds/m). In the ANN model, due to the complex connections between neurons at different layers, a mathematical formula cannot be derived.

3.3. Identification of Dominant Input Variables

In order to determine the relative importance of the input variables included in the best-fit combinations of variables, K-S analysis was conducted and the results for altitude, sand, silt, clay, EC, and pH, are shown in

Table 4. Significance analysis by Kolmogorov–Smirnov test.

Variable	Field Capacity		Permanent Wilting Point
	p-Value	Significance	p-Value	Significance
Altitude	0.704		0.012	*
Sand	0.017	*	0.331
Silt	0.042	*	0.099
Clay	0.042	*	0.331
EC	0.545		0.039	*
pH	-		0.415

* The significance is >0.05.

. The results showed that clay, sand, and silt in the FC model and altitude and EC in the PWP model have significant impacts on preventing higher absolute errors in FC and PWP simulation. Longitude and latitude were not included in the K-S analysis due to the wide range of values for these two parameters in the database. However, it is clear that geographical locations play essential roles in FC and PWP modeling, which agrees with the conclusions of [9,13,23]. Despite the fact that pH and EC do not always decrease the error in FC and PWP modeling, the inclusion of these two parameters in the combination of input variables generally improves the model’s performance.

For the model error distribution analysis, the comprehensive modeling capacity of each model was evaluated by above-mentioned indicators. However, R² only describes the model’s responded variation between dependent variables and independent variables. RMSE, NRMSE, and MAE are averaged errors that cannot represent the error distribution, i.e., indicating the primary range of the error. In order to evaluate each model’s error distribution, the absolute error (AE) was utilized in the testing dataset of each model for both the FC and PWP (Figure 4). Five categories were used, including AE ≤ 1, 1 < AE ≤ 2, 2 < AE ≤ 3, 3 < AE ≤ 4, and AE > 4. Figure 4a shows the AE distribution in the FC model. All linear-based models have lower accuracy (AE ≤ 1%), ranging from 0% to 9.4%, and an unignored error (AE > 4%) from 40.6% to 96.4%. In contrast, ML-based ANN and GEP models demonstrate better simulation ability (AE ≤ 1%) from 25.0% to 37.5% and the massive error (>4%) is from 28.1% to 31.3%. Over 60% of AEs in the three PTFs in previous studies are greater than 4% although the FC modeling result from the study of Adhikary et al. [36] presented an acceptable R² (0.683), but around 78.1% of AEs were higher than 4%.

The result for the PWP simulation was similar to the ANN and GEP models and had greater modeling ability than linear-based models in terms of AE distribution, as shown in Figure 4b. Notably, the GEP model showed a higher simulation ability to avoid large errors (AE > 4%) than the ANN model, almost 90% of AEs in the GEP model were lower than 4% while this value was only 84% in the ANN model. It should be mentioned that the model applied by Adhikary et al. [36] only used sand and silt in modeling the FC and clay in modeling the PWP, meaning that this model has the lowest dataset requirement. In situations in which resources to analyze OM, pH, and EC are not available, equations from Adhikary et al. [36] can be used to provide rough estimations of the FC and PWP values. When EC and pH data are available, the ML-based ANN model is an effective and more accurate algorithm to predict the FC and PWP. However, the ANN model requires more sophisticated hardware support and knowledge than the GEP model. Therefore, considering the applicability and performance, GEP has some advantages over ANN for the FC and PWP simulation.

4. Conclusions

The Field capacity (FC) and permanent wilting point (PWP) are critical metrics in soil health assessment. Yet, direct measurement of their values is often challenging. In the study, data from the World Soil Information Service (WoSIS) were utilized, and advanced algorithms, specifically the artificial neural network (ANN) and gene-expression programming (GEP) algorithms, were employed to predict the FC and PWP using accessible parameters. Optimal variables for modeling the FC and PWP were identified including longitude; latitude; altitude; sand, silt, and clay content; electrical conductivity; and pH. Significantly, the ML-based ANN and GEP models achieved higher modeling accuracy, compared with the conventional, linear-based FC and PWP model developed in Adhikary et al. [36].

The NRMSE evaluation showed that the ANN model was best for PWP prediction, while the GEP model performed best for FC prediction, both providing relatively accurate results. Although greater model interpretability was exhibited by the ANN, greater resilience against substantial errors was demonstrated by the GEP model. However, it is important to note that caution must be exercised when applying these models to current soils because while soil changes are minimal when undisturbed or undeveloped, the collection times for data in the WoSIS database vary. This might result in disparities between the database’s information and the current state of soil development.

Overall, this study revealed that machine learning, particularly in the GEP model, can be effectively used to predict the FC and PWP using WoSIS data. These results can be used to improve rational irrigation practices, akin to IoT-based smart irrigation systems as reported by [54]. This enhanced irrigation scheme could lower water consumption in agriculture, reduce the likelihood of crop failures, and potentially decrease greenhouse gas emissions, marking a significant advancement in soil health evaluation and environmental stewardship. Future studies are encouraged to explore a broader range of machine learning models and to refine predictive accuracy further, thereby enhancing the management of soil resources and the resilience of agricultural ecosystems.