A Novel Three-Segment Model to Describe the Entire Soil–Water Characteristic Curve

: This study aims to accurately describe the soil–water characteristic curve (SWCC) across the full range from saturation to oven dryness. We propose a smooth, continuous three-segmented SWCC model that divides the saturation range into wet, air-dried, and oven-dried segments. The two model junction points are anchored at matric suctions of 10 4.5 and 10 6.5 cm, respectively. The soil water content at 10 4.5 cm represents the maximum soil hygroscopy, reflecting the maximum water content in air-dried soil, while the soil water content at 10 6.5 cm characterizes the minimum soil water content. This imbues the junction points with specific physical significance regarding soil moisture content and matric potential. The model was tested with the water retention data of nine soils across the SWCC and compared with three existing SWCC models based on the adjusted coefficient of determination (adjR 2 ) and root mean square error (RMSE). The results indicated that the proposed model accurately described the entire SWCC. The three-segmented model yielded an adjR 2 of >0.99 and an RMSE of ≤ 0.022 cm 3 cm − 3 , outperforming other models. We also introduce a new method for predicting soil water data in air-dried and oven-dried segments. The results showed that the predicted soil water content values were accurate.


Introduction
The soil-water characteristic curve (SWCC) is a critical property for simulating unsaturated soil behaviors, as it relates the soil water content (θ) to the matric suction (h) [1][2][3].Precise SWCC models are crucial for offering authentic portrayals of soil hydrology, which is fundamental for various applications in agriculture, environmental science, and geotechnical engineering [4,5].
Over the preceding decades, a multitude of SWCC models have emerged to effectively capture retention data [1][2][3]6].Among these, the Brook and Corey (1964) [1] and van Genuchten (1980) [3] models stand out for their widespread adoption in fitting SWCC data within the wet range [7].The term "wet range" typically refers to 0 ≤ h ≤ 10 4.18 (15,000) cm, as 10 4.18 cm corresponds to the permanent wilting point (θ PWP ) [8].These models can be integrated with hydraulic conductivity models [9,10], yet they exhibit limitations when it comes to dry soil conditions [11,12].This limitation is particularly evident when these models are used in numerical simulations, where they fail to describe evaporation from dry soil due to their inability to consider water contents below the residual water content [13,14].Here, "dry soil" typically refers to θ within the "dry range", which spans from θ PWP to Agronomy 2024, 14, 707 2 of 14 the oven-dried soil water content [15,16].The corresponding h value for oven-dried soil is usually 10 7.0 cm [13,17,18].
In response to this challenge, researchers have proposed models for the entire range of soil water content, initially by extending traditional models towards the oven-dried end and later by introducing new models [13,18,19] and novel segmented models [11,16,[19][20][21][22][23].For example, Rossi and Nimmo (1994) [15] introduced a model that splits the SWCC into two or three parts corresponding to the capillary flow and film flow states by defining a critical pressure head and critical water content.In addition, Du (2020) [16] proposed a novel segmented model that divides the SWCC into three stages of saturated flow, capillary flow, and film flow, providing a continuous and smooth description of the full moisture range from saturation to drying.
These models aim to provide a more comprehensive description of the SWCC from saturation to oven dryness, acknowledging the distinct characteristics of the high, medium, and low suction ranges of SWCCs [23,24].According to Du's (2020) study [16], the threesegmented model better describes the entire SWCC, from saturation to oven dryness, than other models.However, the two connection points in the three-segmented model are not fixed, and thus, the accuracy of the model as a whole is influenced by these connection points [16].For example, regarding the h value corresponding to the connection point (h c ) between the second and third segments of the model, the larger the h value (when 10 3.0 ≤ h c ≤ 10 6.0 cm), the better the overall accuracy of the model, with a higher adjusted determination coefficient (adjR 2 ) and a smaller root mean square error (RMSE).
Thus, we believe it is necessary to assign a specific physical meaning to h at the connection points in a three-segmented model that covers the entire saturation range, making them fixed values rather than adjustable fitting parameters.Additionally, in laboratory settings, the SWCCs for numerous soils are typically assessed under wet conditions, with their maximum h values falling between 10 3.0 and 10 4.18 cm, because determining the SWCC from the permanent wilting point down to the oven-dried soil water content can occasionally be challenging and labor intensive.Therefore, it is also necessary to develop methods for predicting soil water data (SWD) in the high suction range (e.g., h > 10 4.18 cm), addressing the issue of the time-consuming and labor-intensive nature of measuring SWD in this range.
The objectives of this study encompass the following: 1  ⃝ providing a novel model for depicting the SWCC; 2  ⃝ further developing and refining the segmented model to endow the matric suction or moisture content at the connection points with distinctive physical significance, aiming for an accurate description of the SWCC; and 3  ⃝ predicting the SWD for h > 10 4.18 cm.

Description of the SWCC
In this study, the SWCC from saturation to oven dryness is divided into three segments.The first stage is the wet segment, where the variation range of θ is from θ S to θ MH .Here, θ S represents the soil saturated water content, and θ MH represents the maximum hygroscopy of the soil, which represents the maximum θ of air-dried soil.The second stage is the airdried segment, where the θ varies between θ MH and θ ADC .In this study, θ ADC is referred to as the air-dried coefficient of soil, which characterizes the minimum θ of air-dried soil.The third stage is the oven-dried segment, corresponding to θ within the interval where 0 ≤ θ ≤ θ ADC , indicating the soil water that can only be removed from the soil through drying methods.
It is essential to assign values to h corresponding to θ MH and θ ADC , giving them clear physical significance and explaining the rationale behind it.The h corresponding to θ MH (h MH ) is considered to be 10 4.5 cm [25], which can be calculated using the fundamental Clausius-Clapeyron equation as follows [26]: where p = {Q/Tr − Rln(fr)/M}, Q = 2401 ± 3 kJ kg −1 is the specific heat of evaporation for the temperature range of 0-105 • C, R = 8.314 J mol −1 K −1 is the universal gas constant, T [K] is the absolute temperature for a thermodynamic reservoir (laboratory) with constant relative air humidity (fr) at room temperature (Tr), and M = 0.018 kg mol −1 is the molar mass of water.By scaling the matric suction Ψz with the density of water (ρ w = 1000 kg m −3 ) and transforming the equivalent pressure from Pascals to a water head height in centimeters (101.3 kPa equates to 1030 cm), the following formula for estimating h is derived: For θ MH , the value of fr equals 0.98 [27] and T = Tr ranges from 15 to 25 • .Employing Equations ( 1) and ( 2) for the estimation yields a fluctuation in h, ranging from 10 4.44 to 10 4.45 cm.
h MH also can be calculated using the formula provided by Schofield (1935) [28]: where H r is the relative humidity in the soil atmosphere (expressed as a percentage).The value of fr corresponding to θ MH equals 0.98 [27].Thus, the H r value corresponding to θ MH equals 98, resulting in pF = 4.54.The relationship between h and pF is given by the following equation: h = 10 pF (4) Thus, h MH = 10 4.54 cm, and the h MH range of 10 4.4 cm to 10 4.54 cm is feasible.Consequently, h MH varies from 10 4.44 cm to 10 4.54 cm, with an approximate value of 10 4.5 cm.
In this study, the fr corresponding to θ ADC is regarded as 0.10.Thus, when T = Tr ranges from 15 to 25 • C, the h corresponding to θ ADC (h ADC ) varies between 10 6.49 cm and 10 6.51 cm.Based on Equation (3), the pF corresponding to θ ADC is 6.6 because Hr = 10.Thus, h ADC is equal to 10 6.6 cm as calculated using Equation (4).Therefore, h ADC ranges from 10 6.49 cm to 10 6.6 cm, and the approximate value of 10 6.5 cm is used as the h ADC in this study.

Three-Segmented SWCC Model
The SWCC model at the wet segment is expressed as follows: where θ (cm 3 cm −3 ) represents the soil water content, θ S (cm 3 cm −3 ) represents the saturated soil water content, θ R (cm 3 cm −3 ) represents the residual water content, h (cm) represents the matric suction, h a (cm) represents the air-entry suction, m is an adjustable fitting parameter, and h MH (cm) is the matric suction corresponding to θ MH .θ MH refers to the greatest amount of water that the soil can retain under the conditions of 98% relative humidity [27].It can be considered as the maximum θ value of air-dried soil.The maximum value of Equation ( 5) is anchored at θ S , and the minimum value is anchored at θ MH .Therefore, Equation ( 5) is referred to as the wet-segment model of the SWCC in this study.
In Equation (5), the second derivative of the θ(h) vs. ln(h) function, denoted as S ′ , can be expressed as the following: When Equation (6) equals zero, the value of h obtained from solving it represents the matric suction at the inflection point of SWCC, expressed as the following: Substituting Equation (7) into Equation ( 5), we obtain the following expression for the soil water content at the inflection point: The expression of Equation ( 7) can be transformed into Equation (9), which states the following: M = h a /h i (9) This equation clearly expresses the physical meaning of the parameter m, which is the ratio of the air entry suction (h a ) to the suction at the inflection point (h i ) of Equation (5).The θ corresponding to h a is considered equal to θ s [1,6].Theoretically, θ i should be less than θ s , which would imply that h i should be greater than h a .Therefore, m is less than 1.
At the air-dried segment, the SWCC is expressed as follows: where θ MH (cm 3 cm −3 ) represents the maximum hygroscopy of soil, h MH (cm) is the matric suction at θ MH , h ADC = 10 6.5 cm is the matric suction corresponding to θ ADC , and ω is an adjustable fitting parameter.Equation ( 10) is anchored at θ MH and θ ADC , where θ MH and θ ADC characterize the maximum and minimum θ of air-dried soil, respectively.Therefore, in this study, Equation ( 10) is referred to as the air-dried segment model of the SWCC.
In Equation (10), the second derivative of the function relating θ(h) to ln(h), represented by S ′ , can be formulated as follows: Upon setting Equation (11) to zero, the resulting value of h from its solution corresponds to the matric suction at the inflection point of the SWCC, which is denoted as follows: where h INF (cm) the is matric suction at the inflection point of Equation (11).By inserting Equation (12) into Equation (10), the formula for the soil water content at the inflection point can be derived as follows: The formulation of Equation ( 12) can be rearranged to yield Equation ( 14), indicating the following: Equation ( 14) explicitly defines the physical significance of the parameter ω, which is the ratio of h MH to h I .In theory, θ INF is expected to be lower than θ MH , suggesting that h I should exceed h MH .Consequently, ω is inferred to be less than 1.
For the oven-dried segment, the SWCC model is written as follows: where h Z (cm) is the matric suction at the oven-dried soil water content (θ Z ), typically ranging from 10 6.85 to 10 7.0 cm [13,17,18], and λ is an adjustable fitting parameter.Equation ( 15) accurately anchors at θ ADC and θ Z = 0.
The measured θ is calculated as follows: where θ (cm 3 cm −3 ) is the soil water content, W (g) is the mass of soil, W d (g) is the mass of the oven-dried soil, D b (g cm −3 ) is the soil bulk density, and ρ w (g cm −3 ) is the water density.For oven-dried soil, W equals W d , hence θ Z is equal to zero.

Predicting SWD within the Interval
Where 0 ≤ θ ≤ θ PWP Due to the time and effort involved in determining the SWD in the high suction range, laboratory measurements typically cover the h range of 0-10 4.18 cm, corresponding to the soil moisture range of θ S -θ PWP .Therefore, it would be highly meaningful to predict the data for h > 10 4.18 cm using SWD for h < 10 4.18 cm.Previous studies [16,24] divided soil saturation conditions into three parts: saturated flow, capillary flow, and film flow.The value h = 10 3.75 cm marks the boundary of capillary flow and characterizes the maximum value of film flow.Hence, an attempt can be made to fit the SWCC using SWD at 10 3.75 cm, 10 4.0 cm, 10 4.18 cm, and 10 7.0 cm for predicting data in the range of 10 4.18 ≤ h < 10 7.0 cm, which covers the air-dried and oven-dried segments of the SWCC in this study.Here, h = 10 7.0 cm corresponds to zero water content and can be considered as theoretically known data that does not require measurement.The SWCC can be fitted using Equation ( 18): where h e , γ, θ c , and θ d are parameters.It is important to note that Equation ( 18) is similar to Equation ( 5), but Equation ( 18) aims to predict the SWD for 10 4.18 < h < 10 7.0 cm, and thus, its parameters may not have specific physical meanings.

Experimental Data
The SWD set consisting of nine soils extracted from Koorevaar et al. (1983) [17] was used in this work.The SWCC was originally plotted as θ vs. pF by Koorevaar et al. (1983), with all curves ending at pF = 7.0 at zero soil water content.Values of θ were extracted at intervals of pF = 0.25 in the range of pF values from 0 to 7.0.Values of θ corresponding to the permanent wilting point (θ PWP ) also were extracted at pF = 4.18.Values of h were calculated using the formula h = 10 pF , and θs was estimated using θ at pF = 0.

Fitting the Entire SWCC
The three-segmented SWCC was fitted using SPSS 19.0.Equations ( 5), (10) and (15) were fitted one by one, carrying the parameters from one to the next.To achieve smooth and seamless connection of the three-segmented curve, it was ensured that the calculated θ MH and θ ADC from Equations ( 5) and (15), respectively, matched the measured values of θ MH and θ ADC .

Statistical Analyses
To evaluate the performance of the SWCC model, two key metrics were employed: the adjusted coefficient of determination (adjR 2 ) and the RMSE.
The adjR 2 is determined using the following formula: where R 2 represents the determination coefficient; n is the total number of data points for each soil sample in the SWCC dataset; j is the number of parameters in the SWCC model; θ m and θ p denote the observed and fitted θ, respectively; and θ k represents the mean of the θ m values.
The RMSE is computed using the following formula: These statistical measures offer significant insights into the SWCC model's effectiveness and precision.The adjR 2 considers both the model's fit quality and the complexity indicated by the number of parameters, whereas the RMSE measures the aggregate discrepancy in model predictions.Through the evaluation of these indicators, we can efficiently determine the appropriateness of the SWCC model for our particular SWD set.

Test of the Three-Segmented Model for the Entire SWCC
Figure 1 illustrates the observed data and the fitted three-segmented models of the entire SWCC in the semilog coordinate system for nine different soils.In the figure, solid dots represent the observed SWD, while the solid black lines represent the fitted threesegmented model of the entire SWCC.The three-segmented model provides continuous and smooth curves from saturation to oven dryness by satisfactorily fitting the SWD for the nine soils.As shown in Figure 1, the fitted models of the entire SWCC closely match the SWD for all nine soils.For the calcareous loam, silt loam, and young oligotrophic peat soil samples, the goodness of fit values of the three-segmented SWCC models are almost the same.Thus, the adjR 2 values of calcareous loam, silt loam, and young oligotrophic peat are 0.998, 0.998, and 0.999, respectively, and the RMSE values are 0.007, 0.008, and 0.013 cm 3 cm −3 , respectively.For the other six soil retention datasets, the fitting performances of the three-segmented model are also good.As Table 1 presents, the adjR 2 values of the nine soils are notably high, ranging from 0.991 to 0.999.Furthermore, their RMSE values are comparatively low, with a range of 0.007 to 0.022 cm 3 cm −3 .As can be seen from Figure 1, the fitting SWCCs for dune sand and loamy sand exhibit a better match with the SWD in the range of 10 4.5 ≤ h ≤ 10 7.0 cm than in the range of 0 ≤ h ≤ 10 4.5 cm.Thus, it was necessary to further analyze the accuracy of the wet end model, air-dried end model, and oven-dried end model.The adjR 2 and RMSE for these three models were therefore calculated separately.Figure 2 displays the adjR 2 and RMSE for the wet segment model and air-dried segment model of the nine soils.As shown in Figure 2a,b, the adjR 2 values of the wet segment model for the nine soils varied between 0.944 and 0.999, and the RMSE values ranged from 0.005 to 0.026 cm 3 cm⁻ 3 .As illustrated in Figure 2c,d, it was evident that the air-dried segment models for the nine soils exhibited a range As can be seen from Figure 1, the fitting SWCCs for dune sand and loamy sand exhibit a better match with the SWD in the range of 10 4.5 ≤ h ≤ 10 7.0 cm than in the range of 0 ≤ h ≤ 10 4.5 cm.Thus, it was necessary to further analyze the accuracy of the wet end model, air-dried end model, and oven-dried end model.The adjR 2 and RMSE for these three models were therefore calculated separately.Figure 2 displays the adjR 2 and RMSE for the wet segment model and air-dried segment model of the nine soils.As shown in Figure 2a,b, the adjR 2 values of the wet segment model for the nine soils varied between 0.944 and 0.999, and the RMSE values ranged from 0.005 to 0.026 cm 3 cm −3 .As illustrated in Figure 2c,d, it was evident that the air-dried segment models for the nine soils exhibited a range of adjR 2 values between 0.967 and 0.995, with RMSE values varying between 0.001 and 0.014 cm 3 cm −3 .This indicates that the air-dried segment model provides a better description of the SWCC in the range of 10 4.5 ≤ h ≤ 10 6.5 than the wet segment model does in the range of 0 ≤ h ≤ 10 4.5 .
The oven-dried segment models for the nine soil samples had adjR 2 = 1 and RMSE = 0.This is because the oven-dried segment SWCC in this study only includes data points at 10 6.5 , 10 6.75 , and 10 7.0 cm.According to the oven-dried segment model, the precise anchoring points are at h ADC = 10 6.5 cm (θ ADC ) and h Z = 10 7.0 cm (θ Z = 0).An appropriate adjustable fitting parameter λ value ensures that the fitted model accurately calculates the θ value at h = 10 6.75 cm, which is in exact agreement with the measured value.Therefore, the fitted values of θ corresponding to h = 10 6.5 , 10 6.75 , and 10 7.0 cm are exactly equal to the observed values.Thus, the fitted SWCC closely matches the observed SWD within the range of 10 6.5 ≤ h ≤ 10 7.0 cm. of adjR 2 values between 0.967 and 0.995, with RMSE values varying between 0.001 and 0.014 cm 3 cm −3 .This indicates that the air-dried segment model provides a better description of the SWCC in the range of 10 4.5 ≤ h ≤ 10 6.5 than the wet segment model does in the range of 0 ≤ h ≤ 10 4.5 .The oven-dried segment models for the nine soil samples had adjR 2 = 1 and RMSE = 0.This is because the oven-dried segment SWCC in this study only includes data points at 10 6.5 , 10 6.75 , and 10 7.0 cm.According to the oven-dried segment model, the precise anchoring points are at hADC = 10 6.5 cm (θADC) and hZ = 10 7.0 cm (θZ = 0).An appropriate adjustable fitting parameter λ value ensures that the fitted model accurately calculates the θ value at h = 10 6.75 cm, which is in exact agreement with the measured value.Therefore, the fitted values of θ corresponding to h = 10 6.5 , 10 6.75 , and 10 7.0 cm are exactly equal to the observed values.Thus, the fitted SWCC closely matches the observed SWD within the range of 10 6.5 ≤ h ≤ 10 7.0 cm.

Comparison of SWCC Models
To evaluate the behavior of the three-segmented model proposed in this study, we compared it with three additional popular SWCC models.The expressions and parameters of the seven SWCC models are listed in Table 2.The Brooks-Corey (BC) model [1] and van Genuchten (vG) model [3] are traditional models for describing the SWCC between θS and θR and are very popular.The Du model [16] is a three-segmented model, which is superior to other segmented models such as the Rossi and Nimmo model [15], the Peter model, the Zhang model [23], the DG model [29], and the Webb model [11], based on the research of Du [16].

Comparison of SWCC Models
To evaluate the behavior of the three-segmented model proposed in this study, we compared it with three additional popular SWCC models.The expressions and parameters of the seven SWCC models are listed in Table 2.The Brooks-Corey (BC) model [1] and van Genuchten (vG) model [3] are traditional models for describing the SWCC between θ S and θ R and are very popular.The Du model [16] is a three-segmented model, which is superior to other segmented models such as the Rossi and Nimmo model [15], the Peter model, the Zhang model [23], the DG model [29], and the Webb model [11], based on the research of Du [16].

SWCC Model
Function Parameters Reference Du model As shown in Figure 3, the adjR 2 and RMSE were employed to evaluate and compare the performances of the three-segmented model and the three other SWCC models.Although the maximum adjR 2 value was close to 1 and the 25th percentile was greater than 0.80 for each model (Figure 3a), the variations in the adjR 2 values of the models were different.The adjR 2 for the three-segmented model only varied between 0.99 and 1.00.The 25th and 75th percentiles of the adjR 2 values for the Du model were similar to those of the three-segmented model, but the range of the minimum to the maximum adjR 2 values of the Du model was greater than that of the three-segmented model.The BC model, which is a traditional single-segment model, had the largest adjR 2 range, with a minimum of 0.769 and a maximum of 0.999.The adjR 2 values of the vG model ranged from 0.805 to 0.999.The results indicate that the three-segmented model proposed in this study slightly surpasses the Du model in describing the observed soil water retention data, and that it explains these data more effectively than single-segment models such as the BC and the vG models.
hough the maximum adjR 2 value was close to 1 and the 25th percentile was greater than 0.80 for each model (Figure 3a), the variations in the adjR 2 values of the models were different.The adjR 2 for the three-segmented model only varied between 0.99 and 1.00.The 25th and 75th percentiles of the adjR 2 values for the Du model were similar to those of the three-segmented model, but the range of the minimum to the maximum adjR 2 values of the Du model was greater than that of the three-segmented model.The BC model, which is a traditional single-segment model, had the largest adjR 2 range, with a minimum of 0.769 and a maximum of 0.999.The adjR 2 values of the vG model ranged from 0.805 to 0.999.The results indicate that the three-segmented model proposed in this study slightly surpasses the Du model in describing the observed soil water retention data, and that it explains these data more effectively than single-segment models such as the BC and the vG models.Figure 3b illustrates the RMSE variations of the four SWCC models.The minimum RMSE value of each model was close to 0, but the maximum varied greatly.The maximum RMSE values for the three-segmented model proposed here and the BC, vG, and Du models were 0.022, 0.086, 0.073, and 0.033 cm 3 cm −3 , respectively.The 75th percentiles of these four models were not greater than 0.08.Overall, the three-segmented model proposed in this study outperforms other models with a significantly smaller error.Figure 3b illustrates the RMSE variations of the four SWCC models.The minimum RMSE value of each model was close to 0, but the maximum varied greatly.The maximum RMSE values for the three-segmented model proposed here and the BC, vG, and Du models were 0.022, 0.086, 0.073, and 0.033 cm 3 cm −3 , respectively.The 75th percentiles of these four models were not greater than 0.08.Overall, the three-segmented model proposed in this study outperforms other models with a significantly smaller error.

Predicting the Soil
Water Content within the Interval of 0 ≤ θ ≤ θ PWP Equation (18) was fitted using SWD from the following four points: h = 10 3.75 , 10 4.0 , 10 4.18 , and 10 7.0 cm.The predicted values of θ in the matric suction range of 10 4.18 ≤ h ≤ 10 7.0 cm, as estimated using Equation (18), were plotted against the measured values in a linear relationship in Figure 3. Highly significant linear relationships were found between the predicted and measured θ values (Figure 4).In theory, if the slope of the linear equation is equal to 1.0, the y-intercept is equal to zero, and the adjR 2 is equal to 1, the predicted values should match the measured values exactly.As shown in Figure 4, the slopes of the linear equations for the nine soils ranged from 0.938 to 1.019, and the y-intercepts varied between −0.0102 and 0.0164, with adjR 2 values ranging from 0.993 to 0.997.This indicates that the predicted θ values from Equation ( 18) closely match the measured θ values.Therefore, it is feasible to use the method provided in this study to predict SWD in the range of 10 4.18 ≤ h ≤ 10 7.0 cm.linear equations for the nine soils ranged from 0.938 to 1.019, and the y-intercepts varied between −0.0102 and 0.0164, with adjR 2 values ranging from 0.993 to 0.997.This indicates that the predicted θ values from Equation ( 18) closely match the measured θ values.Therefore, it is feasible to use the method provided in this study to predict SWD in the range of 10 4.18 ≤ h ≤ 10 7.0 cm.Relationships between the measured soil water content and the soil water content predicted using Equation (18) proposed in this study.The dots in the figure represent the actual measured values of soil water content, the solid lines indicate the linear regression between the measured soil water content and the soil water content predicted using Equation (18), and the dashed lines represent the prediction intervals.
To evaluate the prediction obtained using Equation ( 18), we compared it with the logarithmic model proposed by Du (2020) [16].The logarithmic model is expressed as follows: where θ (cm 3 cm −3 ) is the soil water content, h (cm) is the matric suction, hZ (cm) is the matric suction of oven-dried soil, and θb (cm 3 cm −3 ) and hb (cm) are adjustable parameters.To evaluate the prediction obtained using Equation ( 18), we compared it with the logarithmic model proposed by Du (2020) [16].The logarithmic model is expressed as follows: where θ (cm 3 cm −3 ) is the soil water content, h (cm) is the matric suction, h Z (cm) is the matric suction of oven-dried soil, and θ b (cm 3 cm −3 ) and h b (cm) are adjustable parameters.
According to the SWD in this study, the h Z (cm) is equal to 10 7.0 cm, and θ b = θ PWP corresponds to h b = 10 4.18 cm.A strong linear correlation was identified between the estimated and actual θ values, as depicted in Figure 5.The slopes of the linear regression equations for the nine soils spanned from 0.9314 to 1.001, while the y-intercepts fluctuated from 0.0014 to 0.0293 (Figure 5).The adjR 2 values shown in Figure 5 vary between 0.849 and 0.988.
in Figures 4 and 5 reveals that Equation ( 18) exhibits a narrower range of adjR values tighter range of slopes, and smaller y-intercepts compared to Equation ( 22).Considerin these points, it is reasonable to conclude that Equation ( 18) offers enhanced predicti accuracy for the soil water content within the interval of 10 4.18 < h < 10 7.0 cm than Equatio (22).It is crucial to note that while Equation ( 18) exhibits a high degree of predictive acc racy, it lacks inherent physical interpretability.This is because the fitting parameters Equation ( 18), as listed in Table 3, lack physical meaning.For example, the parameter is less than zero, and θc is greater than 1.0.Therefore, Equation ( 18) can only be used predict θ in the range of 10 4.18 ≤ h ≤ 10 7.0 cm and cannot be used to describe the SWCC.practical applications, Equation ( 18) is initially employed to predict SWD within the ran of 10 4.18 < h < 10 7.0 cm.Subsequently, these predicted SWD can be used to fit the SWC with the air-dried and oven-dried segment models.A comparative analysis of the slopes, y-intercept values, and adjR 2 values presented in Figures 4 and 5 reveals that Equation (18) exhibits a narrower range of adjR 2 values, a tighter range of slopes, and smaller y-intercepts compared to Equation (22).Considering these points, it is reasonable to conclude that Equation ( 18) offers enhanced predictive accuracy for the soil water content within the interval of 10 4.18 < h < 10 7.0 cm than Equation ( 22).
It is crucial to note that while Equation ( 18) exhibits a high degree of predictive accuracy, it lacks inherent physical interpretability.This is because the fitting parameters of Equation ( 18), as listed in Table 3, lack physical meaning.For example, the parameter θ d is less than zero, and θc is greater than 1.0.Therefore, Equation ( 18) can only be used to predict θ in the range of 10 4.18 ≤ h ≤ 10 7.0 cm and cannot be used to describe the SWCC.In practical applications, Equation ( 18) is initially employed to predict SWD within the range of 10 4.18 < h < 10 7.0 cm.Subsequently, these predicted SWD can be used to fit the SWCC with the air-dried and oven-dried segment models.

Discussion
The entire SWCC describes the SWD across the full range from saturation to oven dryness [13,[16][17][18].The h corresponding to oven-dried soil is referred to as 10 7.0 cm in this study.This value is reasonable based on the calculations of Equations ( 1) and ( 2).For oven-dried soil, T = 378 K (105 • C) and 288K ≤ Tr ≤ 298K and 0.2 ≤ f ≤ 0.7.The calculation using Equations ( 1) and ( 2) yields an approximate value of h as 10 7.0 cm [18].The θ value of oven-dried soil (θ Z ) is considered zero.This does not imply that there is no water in the oven-dried soil, but rather indicates that the water remaining in the soil after equilibrium at 105 • C has not been expelled.Consequently, at this point, the θ calculated according to Equation ( 16) is equal to zero.
θ MH is referred to as the maximum θ of air-dried soil, which characterizes the boundary between the wet segment model and the air-dried segment model.θ MH is defined as the maximum amount of water that air-dried soil particles can absorb from an environment close to water vapor saturation [25].Thus, it is reasonable to use θ MH as the maximum θ of air-dried soil.
The dividing point between the air-dried segment and the oven-dried segment is anchored at θ ADC , corresponding to h = 10 6.5 cm, which describes the minimum θ value of air-dried soil.The relative air humidity corresponding to θ ADC is 10% (0.1) at room temperature.Generally, it is difficult for the relative air humidity to drop to 10% at room temperature.Therefore, defining θ ADC as the minimum θ of air-dried soil is theoretically reasonable, as the θ of air-dried soil usually exceeds θ ADC , corresponding to h less than 10 6.5 cm.
The SWCC encompasses both the water adsorption and water release functions.In this study, the two anchor points of the three-segmented model are fixed at θ MH and θ ADC , which represent the maximum and minimum θ values of air-dried soil, respectively.Consequently, the three-segmented model of the SWCC proposed in this research falls within the category of the water release function.

Conclusions
By introducing segmented models for the wet, air-dried, and oven-dried segments, this study successfully describes the SWCCs from saturation to oven dryness.These models not only independently describe the soil moisture characteristics within their respective segments but also provide a continuous and accurate description of the entire moisture range when combined.The adjR 2 values of the three-segmented models for nine soils were consistently above 0.991, and the RMSE values did not exceed 0.022 cm 3 cm −3 .
The connection point between the wet segment model and the air-dried segment model is anchored at h = 10 4.5 cm, corresponding to θ MH .θ MH has a clear physical significance and is widely used in soil moisture research.Therefore, the connection point between the wet segment and air-dried segment models is not only an adjustable parameter but has a definite physical meaning.The connection point between the air-dried segment model and the oven-dried segment model is anchored at h = 10 6.5 cm for θ ADC .θ ADC is considered to represent the water content of air-dried soil at a relative humidity of 10% at room temperature.This perspective is proposed in this study and carries a degree of arbitrariness.Thus, further research is needed in the future to validate the feasibility of θ ADC as a theoretical value for the minimum water content of air-dried soil.Overall, the first and second junction points of the three-segmented model for the entire SWCC are anchored at θ MH and θ ADC , respectively, providing new insights for understanding and applying SWCCs.
The prediction methods proposed in this research show high accuracy in predicting the soil moisture content in the high suction range, which is significant for reducing experimental workloads and improving work efficiency.This provides a new means for obtaining SWD in the high suction range.

Figure 1 .
Figure 1.Fitting results of three-segmented model of the entire soil-water characteristic curve (SWCC) for nine soils.

Figure 1 .
Figure 1.Fitting results of three-segmented model of the entire soil-water characteristic curve (SWCC) for nine soils.

Figure 2 .
Figure 2. Values of the adjusted coefficient of determination (adjR 2 ) and root mean square error (RMSE) of the wet segment model (a,b) and the air-dried segment model((c,d) for nine soils.

Figure 2 .
Figure 2. Values of the adjusted coefficient of determination (adjR 2 ) and root mean square error (RMSE) of the wet segment model (a,b) and the air-dried segment model (c,d) for nine soils.

Figure 3 .
Figure 3.Comparison of the three-segmented SWCC model and other segmented SWCC models in terms of adjR 2 (a) and RMSE (b).The horizontal solid line in each box signifies the median value, and the bottom and top of the box represent the 25th and 75th percentiles, respectively.The error bars represent the range from the minimum to the maximum values.Thr (three-segmented model proposed in this study), BC (Brooks-Corey model), vG (van Genuchten model), and Du (three-segmented model proposed by Du, 2020 [16]).

Figure 3 .
Figure 3.Comparison of the three-segmented SWCC model and other segmented SWCC models in terms of adjR 2 (a) and RMSE (b).The horizontal solid line in each box signifies the median value, and the bottom and top of the box represent the 25th and 75th percentiles, respectively.The error bars represent the range from the minimum to the maximum values.Thr (three-segmented model proposed in this study), BC (Brooks-Corey model), vG (van Genuchten model), and Du (three-segmented model proposed by Du, 2020 [16]).

Figure 4 .
Figure 4. Relationships between the measured soil water content and the soil water content predicted using Equation(18) proposed in this study.The dots in the figure represent the actual measured values of soil water content, the solid lines indicate the linear regression between the measured soil water content and the soil water content predicted using Equation(18), and the dashed lines represent the prediction intervals.

Figure 4 .
Figure 4. Relationships between the measured soil water content and the soil water content predicted using Equation(18) proposed in this study.The dots in the figure represent the actual measured values of soil water content, the solid lines indicate the linear regression between the measured soil water content and the soil water content predicted using Equation(18), and the dashed lines represent the prediction intervals.

Figure 5 .
Figure 5. Relationships between the measured soil water content and the soil water content p dicted using the logarithmic model proposed by Du (2020) [16].The dots in the figure represent t actual measured values of soil water content, the solid lines indicate the linear regression betwe the measured soil water content and the soil water content predicted using the logarithmic mod proposed by Du (2020) [16], and the dashed lines represent the prediction intervals.

Figure 5 .
Figure 5. Relationships between the measured soil water content and the soil water content predicted using the logarithmic model proposed by Du (2020) [16].The dots in the figure represent the actual measured values of soil water content, the solid lines indicate the linear regression between the measured soil water content and the soil water content predicted using the logarithmic model proposed by Du (2020) [16], and the dashed lines represent the prediction intervals.

Table 1 .
Fitting results of three-segmented model of the entire SWCC for nine soils.

Table 1 .
Fitting results of three-segmented model of the entire SWCC for nine soils.

Table 2 .
Three SWCC models for comparison.

Table 2 .
Three SWCC models for comparison.