Efficient Estimation of the Number of Water Retention Curves Required for Applying a Scaling Technique to the Forest Soil

Abstract

For the numerical simulation of rainwater infiltration in forest slope, information on the water retention curve (WRC), which shows spatial variability due to the forest ecosystem and weathered granite in natural forest soils, is required. A scaling approach using three parameters of the LN model has been developed to simplify the spatial variability in the WRCs of the forest slope of the soil under the geomorphological process. This approach showed that we required a spatial data set in scaling parameter, effective porosity, θ_e, and each average value of the remaining two parameters (the matric pressure head corresponding to the median pore radius, ψ_m, and the width of the pore-size distribution, σ), which were defined as reference parameters. In this study, we estimated the minimum number of WRCs required to determine the reference parameters effectively. For this purpose, 77 WRCs of core samples were collected from the whole 25 m forest slope, and we randomly sampled WRCs using a Monte Carlo simulation. The effect of scaling (EOS) increased with the sample size, and the increase became small at a sample size of approximately 20. We could explain 78% (EOS = 0.78) of spatial variability in the WRCs at the 95% confidence level by using the reference parameters derived from eight samples. In addition, we performed stratified sampling to reduce the number of WRCs required. As a result, the sampling scheme, which considers the variability in only slope direction, was the most advantageous. This result indicated that the geomorphological process, which produces spatial variability in the reference parameters of forested slopes, is an important factor for effectively determining reference parameters. This paper concluded that the scaling approach enables us to reduce the required number of samples for WRCs.

1. Introduction

For climate change, a quantitative assessment of rainwater infiltration into the soil is fundamental for evaluating the slope collapse caused by heavy rainfall in mountainous areas. The rainwater infiltration process was theoretically modeled and numerically simulated using the hydraulic properties of unsaturated soil, which were represented by the water retention curve (WRC) (the relationship between the volumetric soil water content, θ, and matric pressure head, ψ) and by unsaturated hydraulic conductivity. The WRC parameters are a part of the function of unsaturated hydraulic conductivity and are crucially important. These WRCs in the drying path are formed by pore size distribution [1] and show a large spatial variability within a natural forested slope [2,3].

However, it is usually not feasible to directly determine the spatial variability in the WRC from the sampling of soil on the entire forest slope and measurements in the laboratory because these studies are time-, labor-, and cost-intensive. Therefore, the hydraulic properties of unsaturated soils have been assumed to be homogeneous in many studies on the numerical simulation of rainwater infiltration [4,5,6]. However, simulations assuming homogeneous soil cannot fully predict the actual water flow in the field [7,8].

To date, many studies have developed a pedo-transfer function to predict WRC [9,10,11,12]. The pedo-transfer function could determine the WRCs easily. These methods require texture data, bulk density, and organic matter content, which require tiresome measurement and time.

A scaling technique was developed as an effective tool for simplifying the description of spatial variability in WRCs. The original scaling theory was introduced by Miller and Miller (1956) [13], who derived the scaling theory using similar media theory. Scaling simplifications related WRCs at different spatial locations to a parameter called a scaling factor using the WRC of a set of reference parameters. Based on this concept, the microscopic structures of soils are assumed to be identical, and the soils differ only in their microscopic length scale, which is characterized by a scaling factor for each soil sample. This scaling technique is widely used in agricultural fields and grasslands [14,15]. In contrast to soils in agricultural fields and grasslands, few studies have applied this technique to forest soils.

Kosugi and Hopmans (1998) [16] indicated that ψ_m represents the microscopic length of the soil using the lognormal model, the LN model [1], which contains three parameters (effective porosity, θ_e; matric pressure head corresponding to the median pore radius of pore-size distribution, ψ_m; and standard deviation of pore-size distribution, σ). Based on this finding, Hayashi et al. (2009) [17] examined three parameters of the LN model to determine the parameters that characterize the spatial variability of WRCs on a natural forested slope. The results indicated that θ_e, instead of ψ_m, mostly characterizes it. They proposed a new scaling approach that evaluates the spatial variability in WRCs using the spatial variability in θ_e. In this approach, θ_e was set as a scaling parameter that exhibits high spatial variability. At this time, the remaining two parameters showing small spatial variability (ψ_m and σ) were set as reference parameters and were identical within the forested slope.

For the practical application of the scaling technique, a simple method for determining the scaling and reference parameters is needed. For this purpose, Nasta et al. (2009) [18] proposed a simple application of a scaling technique based on a similar media theory. In their methods, WRCs were estimated from soil-particle-size distribution data, whose measurement was much simpler than that of the WRC. They demonstrated that the spatial variability of WRCs can be characterized using the proposed method. To easily apply the scaling approach to forest soils, Hayashi et al. (2009) [17] calculated the scaling parameter θ_e as a difference between the water content observed at saturation and the water content observed at ψ = −1000 cm, which is assumed equal to the residual water content. This method explained approximately 90% of the spatial variability in WRCs on the studied natural forested slope. Thus, the WRCs for the entire matric pressure range are not necessary for determining the scaling parameter.

However, little attention has been paid to methods for determining reference parameters. Kosugi and Hopmans (1998) [16] proposed a method in which the reference parameters are optimized by minimizing the residuals of volumetric water content, θ. Several studies have used this method [5,19,20,21,22]. Deurer et al. (2001) [23] derived the reference parameters from geometric means of their parameters. Chari et al. (2020) [24] used an infiltration curve to determine the reference parameters, and Fred Zhang et al. (2004) [25] estimated the reference parameters at the field scale through the inverse modeling of field experiments. Vishkaee et al. (2014) [26] used reference parameters, assuming that the reference soil is the one that consists of uniform-sized spherical particles.

Most previous studies used a large number of WRCs and needed labor approaches to determine the reference parameters. They did not try to reduce the number of WRCs. However, because the values of the reference parameters are common within a site, it may be possible to determine them with high confidence from a small number of WRCs [25]. In addition, in natural forested slopes, the soil structure characterizing the WRC is affected by forest biological activities, weathering, and geomorphological processes in Japan. Thus, there may be a spatial pattern in the spatial variability of the soil structure. Consequently, by recognizing the spatial pattern of the soil structure, we need to select the optimal sampling location to determine the reference parameters. This optimal sampling design for reference parameters, in combination with a simple determination of the scaling parameter proposed by Hayashi et al. (2009) [17], reduces the required number of WRCs and allows us to save effort in measuring the enormous number of WRCs for describing the spatial variability in WRCs in a forested slope. Few studies have focused on the number of samples required to determine the hydraulic properties. Salemi et al. (2020) [27] estimated an adequate number using statistical analysis. However, they analyzed the saturated hydraulic conductivity and not the water retention curves.

Therefore, the aim of this study is to establish the optimal sampling design for WRCs to identify reference parameters and facilitate the application of the scaling technique based on Hayashi et al. (2009) [17] to forest soil. For this purpose, we statistically examined the accuracy of estimation of the WRCs to clarify how many samples are required to obtain these parameters and to suggest a sampling location that accounts for the effects of spatial distribution characteristics of WRCs in the forest slope.

2. Materials and Methods

2.1. Soil Sampling Location

The soil samples were taken from a natural forested slope (main and middle lower panels of Figure 1) underlain by the granitic bedrock (left upper panel of Figure 1) in the Tanakami Mountain range, Shiga Prefecture, Japan, situated at 34°55′00″ N and 135°58′58″ E. Detailed information on this site is provided by Hayashi et al. (2009) [17].

We collected soil samples from vertically distributed profiles, as shown by N_h in Figure 1. N_h was correlated with the scaling parameter θ_e (Hayashi et al., 2009 [17]), and reference parameters were not so correlated.

There was a small dip (indicated by an arrow in Figure 1) on the soil surface of the middle slope. There were slope regions with a thin soil layer below this dip and a thick soil layer above this dip. Thus, the dip may correspond to the head of an ancient landslide, and it was presumed that the thick soil layer consisted of debris brought by the landslide from the middle slope region. This was also supported by the vertical distribution of penetration resistance, N_h (Figure 1). N_h is the number of blows required for a 10 cm soil depth penetration, which was measured using a cone penetrometer with a 60° bit, cone diameter of 2 cm, weight of 2 kg, and fall distance of 50 cm. For points b–d, the soil in the soil layer remained loose (N_h in the range of 0–20 blows/10 cm); however, the soil hardness on the bedrock increased rapidly and reached 100 blows/10 cm, which was the value observed at the interface between the soil layer and the bedrock. Such vertical distributions of N_h values are characteristic of a soil layer consisting of debris [28]. Points e–h, which had a thin soil layer, were thought to be the residual soil remaining after the landslide. For the soils at points i–o, N_h gradually increased with depth. The soil layer indicating such a vertical distribution of the N_h value was considered unaffected by the landslide [28].

Undisturbed soil samples were collected from 15 points distributed from the downslope (point a) to the upslope (point o) segments to assess the spatial variability in the WRC on this forested slope (Figure 1). The sampling depths were set from the surface layer to the layer above the bedrock. The sampling locations are indicated by white dots in the panels in Figure 1. Soil samples were collected in triplicate at each point and depth. But, only at point j, we collected six samples at the depths of 15, 25, 35, 45, and 55 cm, and three samples at the depths of 2.5 and 75 cm.

2.2. Soil Sampling and Laboratory Experiments

Undisturbed soil samples were collected using thin-walled steel core samplers with a volume of 100 cm³ (5 cm inner diameter and 5.1 cm height). A sampler with a sharp edge was inserted vertically into the soil. Impact energy was applied to the cylinder using a hammer-driven device (DIK-1630; Dike Rika Kogyo, Tokyo, Japan). We followed the method described by Grossman and Reinsch (2002) [29] to collect undisturbed soil samples and ensure sampling with minimum disturbance. The roots and organic material were carefully cut around the sampler during the insertion process.

Soil samples were transferred to aluminum trays in the laboratory. These were slowly saturated by wetting from the bottom over a 24 h period, and then weighed to determine the saturated water content, θ_s. Soil WRCs were then measured using the pressure plate method [30] for a matric pressure head, ψ, of −5, −10, −20, −30, −50, −70, −100, −200, −500, and −1000 cm. After measuring the water content at ψ = −1000 cm, each sample was oven-dried to determine the weight of soil in dry condition. The analyses described below were carried out using the average WRCs of three or six replicate samples (each with a volume of 100 cm³) collected at each point and depth. We analyzed the water retention characteristics of 77 soil samples.

There is a problem of hysteresis. The numerical simulation without hysteresis makes the matric potential head more widely fluctuated, compared with the actual soil. However, because analysis including hysteresis is complicated, it is valid that in this study, water retention curves of drying processes are analyzed using the scaling method.

2.3. Water Retention Model

The LN model [1] is based on the assumption that the soil pore radius distribution obeys a lognormal distribution and expresses the WRC as

$$\mathsf{\theta }=\left({\mathsf{\theta }}_{\mathrm{s}}-{\mathsf{\theta }}_{\mathrm{r}}\right)\mathrm{Q}\left({\displaystyle \frac{\mathrm{l}\mathrm{n}\left(\mathsf{\psi }/{\mathsf{\psi }}_{\mathrm{m}}\right)}{\mathsf{\sigma }}}\right)+{\mathsf{\theta }}_{\mathrm{r}}$$

(1)

where θ_r and θ_s (cm³ cm⁻³) are the residual and saturated water contents, respectively; ψ_m (cm) is the matric pressure head corresponding to the median pore radius; σ represents the width of the pore-size distribution; and Q denotes the complementary normal distribution function, defined as

$$\mathrm{Q}\left(\mathrm{x}\right)={\left(2\mathsf{\pi }\right)}^{-1/2}{\int }_{\mathrm{x}}^{\mathrm{\infty }}\exp \left(-{\mathrm{u}}^2/2\right)\mathrm{d}\mathrm{u}$$

(2)

The difference between θ_s and θ_r (i.e., θ_s − θ_r) is referred to as the effective porosity, θ_e.

A plot of Equation (1) shown in Figure 2 demonstrates that ψ_m is the matric pressure head at the inflection point. The parameter σ is defined as follows.

θ[ln(−ψ_m) − σ] − θ[ln(−ψ_m) + σ] = 0.68 × θ_e

(3)

Equation (3) shows that σ controls the magnitude of the change in θ around the inflection point. The parameter θ_e is defined by θ_s − θ_r, and represents the volumetric pore ratio effective for water storage.

2.4. Fitting of the LN Model

Prior to testing the proposed scaling methods, the observed WRC for each soil sample was fitted using the LN model (Fitting Approach IF; Individually Fitting). For the ith soil sample, the jth calculated water content, ${\theta }_{cal,IF}^{i,j}$, was calculated by applying the following equation using a nonlinear least squares optimization procedure (Wraith & Or, 1998) [31]:

$${\theta }_{\mathrm{c}\mathrm{a}\mathrm{l},\mathrm{I}\mathrm{F}}^{i,j}=\left({\theta }_{s,i}-{\theta }_{r,i}\right)Q\left({\displaystyle \frac{ln\left({\psi }^j/{\psi }_{m,i}\right)}{{\sigma }_i}}\right)+{\theta }_{r,i}$$

(4)

where θ_r,i, θ_s,i, ψ_m,i, and σ_i are the parameters for each soil sample i, and ψ^j is the jth matric pressure head. To apply Equation (4), for each soil sample, i, θ_r,i was fixed at the water content observed at ψ = −1000 cm, and θ_s,i was fixed at the observed saturated water content. As shown in Figure 3, the slope of water retention curves at ψ = −1000 cm was almost 0. Therefore, we assumed θ at ψ = −1000 cm was θ_r.

The parameters ψ_m,i and σ_i were optimized by minimizing the residual sum of squares (RSS), computed from

$$RS{S}_i={\sum }_{j=1}^J{\left({\theta }_{obs}^{i,j}-{\theta }_{cal,IF}^{i,j}\right)}^2$$

(5)

where J is the total number of data points for WRC of each soil sample (=9), ${\theta }_{obs}^{i,j}$ is the jth observed water content for a sample i, and ${\theta }_{cal,IF}^{i,j}$ is the calculated water content corresponding to ${\theta }_{obs}^{i,j}$. The sum of RSS_i of all soil samples (RSS_IF) was calculated as follows:

$$RS{S}_{IF}={\sum }_{i=1}^{I}RS{S}_i$$

(6)

where I is the total number of soil samples (=77). This method provides the best description of the spatial variability in soil WRCs under the conditions assumed in the LN model.

In the Fitting Approach UF (Universal Fitting), a single curve was applied to the entire data set with the LN model to exclude the spatial variability in all the parameters (ψ_m, σ, and θ_e) of the pore-size distribution function, and a single set of parameters (i.e., ${\overline{\mathsf{\psi }}}_{\mathrm{m}}$ and $\overline{\mathsf{\sigma }}$) was optimized using

$${\theta }_{cal,UF}^{i,j}=\left({\overline{\theta }}_s-{\overline{\theta }}_r\right)Q\left({\displaystyle \frac{ln\left(\psi /{\overline{\psi }}_m\right)}{\overline{\sigma }}}\right)+{\overline{\theta }}_r$$

(7)

where ${\theta }_{cal,UF}^{i,j}$ is the jth water content calculated for sample i using the Fitting Approach UF. The ${\theta }_{cal,UF}^{i,j}$ is shown in Figure 3. ${\overline{\mathsf{\theta }}}_{\mathrm{s}}$ and ${\overline{\mathsf{\theta }}}_{\mathrm{r}}$ were fixed to the arithmetic mean of all samples to apply Equation (7). The optimization procedure was conducted by minimizing RSS_UF computed from

$$RS{S}_{UF}={\sum }_{i=1}^I\left[{\sum }_{j=1}^J{\left({\theta }_{obs}^{i,j}-{\theta }_{cal,UF}^{i,j}\right)}^2\right]$$

(8)

2.5. Scaling Method

Hayashi et al. (2009) [17] indicated that, among the three parameters expressing the soil pore-size distribution (ψ_m, σ, and θ_e), θ_e mostly characterizes the spatial variability in WRCs in natural forested slopes. Because θ_e is defined by the difference between θ_s and θ_r, they proposed a scaling approach in which θ_s and θ_r are set as scaling parameters differing for each soil, and the remaining two parameters (ψ_m and σ) are set as reference parameters identical within a study site. According to this scaling technique, WRC can be expressed as

$${\theta }_{cal}^{i,j}=\left({\theta }_{s,i}-{\theta }_{r,i}\right)Q\left({\displaystyle \frac{ln\left(\psi /{\widehat{\psi }}_m\right)}{\widehat{\sigma }}}\right)+{\theta }_{r,i}$$

(9)

where ${\theta }_{cal}^{i,j}$ is the jth water content calculated for sample i using Equation (9). θ_r,i and θ_s,i are the residual and saturated water contents of the soil sample i, respectively. ${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$ are the reference parameters, and are identical for the entire dataset.

Hayashi et al. (2009) [17] suggested a scaling method that determines the scaling parameters θ_r,i and θ_s,i for each sample without using a dataset of WRCs, whose measurement requires enormous effort. In this method, θ_s,i is set to be observed θ_s for each soil, and θ_r,i is set to be the water content observed at ψ = −1000 cm for each soil, which was assumed to be equal to the residual water content θ_r. The reference parameters ${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$ were determined using optimization so that the parameters produced the best description of the entire dataset of WRCs for the entire slope. That is, Equation (9) was applied to the whole dataset, and ${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$ were optimized using minimizing the residual sum of squares computed from

$$RSS={\sum }_{i=1}^I\left[{\sum }_{j=1}^J{\left({\theta }_{obs}^{i,j}-{\theta }_{cal}^{i,j}\right)}^2\right]$$

(10)

Using the whole dataset, I and J were 77 and 9, respectively.

2.6. Determination of Reference Parameters

If the universal reference parameters could be determined from a small number of WRCs without using the dataset of WRCs on the entire slope with acceptable confidence, the effort in describing the spatial variability in WRCs on a forested slope would be reduced tremendously. We conducted a numerical experiment involving random sampling of WRCs by varying the sample size, n, using a Monte Carlo approach to examine the efficiency of the reference parameters derived from a limited number of WRCs. The procedure was as follows.

(1) The first step was randomly selecting non-overlapping n WRC sets of WRCs among the 77 WRC datasets collected from the entire slope.

(2) Reference parameters (${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$) were determined using n datasets of the selected WRCs. For this purpose, Equation (9) was applied to n WRC datasets, and ${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$ were optimized by minimizing the RSS computed from Equation (10). θ_s,i was set to be observed θ_s for each soil, and θ_r,i was set to be the water content observed at ψ = −1000 cm for each soil to apply Equation (9). The I and J in Equation (10) were n and 9, respectively.

(3) WRCs were described using the determined reference parameters. That is, the entire datasets of WRCs for the entire slope were computed by substituting one set of reference parameters (${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$) determined in Procedure 2 into Equation (9). The scaling parameters (difference between θ_s,i and θ_r,i) were set to the observed values for each sample i to apply Equation (9). The RSS of the estimated WRCs was calculated using Equation (10). In this case, I and J as expressed in Equation (10) are n and 9, respectively. The accuracy of the scaling method was evaluated using the RSS values.

Procedures 1–3 were repeated 100,000 times for each assumed n value.

The scaling approach demonstrated in Hayashi et al. (2009) [17] corresponds to the case for a sample size of 77 (i.e., n = 77).

2.7. Efficiency of Scaling Method

Using RSS by Equation (10), the following two indices were computed to evaluate the accuracy of the scaling method. The coefficient of determination, R², was calculated as

$$R^2=1-{\displaystyle \frac{RSS}{{\sum }_{i=1}^I\left[{\sum }_{j=1}^J{\left({\theta }_{obs}^{i,j}-{\overline{\theta }}_{obs}\right)}^2\right]}}$$

(11)

where ${\overline{\theta }}_{obs}$ is the arithmetic mean of the observed volumetric water content chosen in random sampling. R² can be used to evaluate the performance of each scaling method by considering the variation in the observed θ values. In the proposed scaling method, I and J were equal to n and 9, respectively.

In addition to R², considering the possible RSS range derived using the scaling method, the effect of scaling, EOS, was computed according to Hayashi et al. (2009) [17]:

$$EOS={\displaystyle \frac{RS{S}_{UF}-RSS}{RS{S}_{UF}-RS{S}_{IF}}}$$

(12)

Here, RSS_IF represents the residual error when the spatial variability is most accurately expressed under the conditions in which the LN model is assumed. In contrast, RSS_UF represents the error when the spatial variability in the pore-size distribution is not considered. Thus, EOS evaluates the effect of each scaling method by considering the possible range of RSS derived from the proposed scaling method. When the scaling method perfectly explains the spatial variation under the conditions in which the LN model is assumed, EOS is equal to one. In contrast, when the scaling method cannot explain spatial variability, EOS is equal to zero.

2.8. Semivariogram of Reference Parameters

We analyzed the spatial distribution of reference parameters using a semivariogram. The definition of semivariogram goes back to Matheron. The experimental semivariograms of log(−ψ_m) and σ were calculated for the slope and vertical direction to assess the spatial pattern of log(−ψ_m) and σ in detail. The semivariogram function was computed as follows (Goovaerts, 1997) [32]:

$$\gamma \left(h\right)={\displaystyle \frac{1}{2N\left(h\right)}}{\sum }^{N\left(h\right)}(z\left(x_i+h\right)-z\left(x_i\right){)}^2$$

(13)

where γ(h) is the semivariance, N(h) is the number of pairs of variables, z(x_i) is the measured variable at spatial location x_i, z(x_i + h) is the measured variable at spatial location x_i + h.

In general, the interval of the lag distance, h, is set to be constant; however, in the present study, N(h) is set to be constant to eliminate the bias of the number of data points (N(h) was fixed to 20 and 15 for the slope and vertical directions, respectively). In order to plot the γ(h), the arithmetic mean of the lag distance in each class was used to plot (h).

3. Result and Discussion

3.1. Observed Data and Efficiency of Fitting with the LN Model

The 77 observed WRCs are shown in Figure 3. Figure 4 and

Table 1. Reference parameters and residual sum of squares (RSS), coefficient of determination (R²), and effect of scaling (EOS), for Fitting Approaches IF and UF, and scaling method proposed by Hayashi (2009) [17], using the overall 77 water retention curves (WRCs).

Method	Reference Parameters		RSS	R2	EOS
	log(−ψm)	σ
Fitting Approach IF	1.302 a (0.206) b	1.577 a (0.386) b	0.033	0.997	1.000
n = 77	1.329	1.718	0.537	0.952	0.838
Fitting Approach UF	1.340	1.739	3.142	0.718	0.000

^a Mean value for all soil samples. ^b Standard deviations.

show the results of fitting the LN model to the WRCs. The Fitting Approach IF plots comparing the observed and calculated θ values were distributed close to the 1:1 line (Figure 4a), resulting in a very high R² (0.997; Table 1) compared with n = 77 and the Fitting Approach UF (Figure 4b,c). This implied that the LN model performed well for each retention dataset at the study site.

Figure 5 shows spatial variabilities of reference parameters (log(−ψ_m) and σ) for each soil sample derived from the Fitting Approach IF. Table 1 summarizes the arithmetic means and standard deviations. The log(−ψ_m) varied in the vertical direction without any clear trend, whereas there was a trend for the slope position: log(−ψ_m) in the middle and lower slope regions (points b, c, and d) was smaller than the arithmetic mean 1.302 of for all data (Table 1), but in most areas of the upper slope region (Points e–o), log(−ψ_m) was greater than the arithmetic mean value. This might be because there are loose soils with large-sized pores from the middle slope region of the lower slope due to past landslides, as noted in the geomorphological processes in Figure 1. The σ for the slope direction has similar trends to log(−ψ_m); that is, at the lower slope region (points a through d), σ was greater than the arithmetic mean of 1.577 for all data (Table 1), but at the middle and upper slopes region (points e through o), σ was smaller than the arithmetic mean. In the vertical direction, σ varied slightly with depth, particularly at points i–o.

As mentioned above, the values of the two parameters differed between the lower and upper slope regions. Most likely, the soils in the lower part of the slope were composed of debris, and the soils in the upper part of the slope were composed of the material remaining after the landslide (Figure 1). Thus, log(−ψ_m) and σ may be affected by the topographic factor. The same trends of log(−ψ_m) and σ slope direction were observed by Hendrayanto et al. (1999) [2] in forested granite slopes.

The fitted curve for the UF approach is shown in Figure 3. The plots comparing the observed and calculated θ values were widely scattered around the 1:1 line (Figure 4c). Hence, R² for the Fitting Approach UF (0.718) was much smaller than that for the Fitting Approach IF (0.997; Table 1).

The parameter log(−ψ_m) optimized using the Fitting Approach UF was similar to the arithmetic mean of log(−ψ_m) obtained using the Fitting Approach IF, whereas σ obtained using the Fitting Approach UF was slightly greater than that obtained using the Fitting Approach IF (Table 1).

3.2. Effectiveness of Scaling Using the Whole WRCs for Determination of Reference Parameters

This section confirmed the accuracy of the scaling method proposed by Hayashi et al. (2009) [17], in which the reference parameters (${\widehat{\psi }}_m$ and $\widehat{\sigma }$) were derived using the whole 77 data sets of WRCs (i.e., n = 77) in the entire forested slope. This accuracy corresponded to that of the case that produced the highest scaling effectiveness with optimal reference parameter values. As shown in Figure 4b, the plots for a sample size of 77 comparing the observed and calculated θ values were distributed close to the 1:1 line with a high R² (0.952; Table 1). EOS was 0.838, and this result indicated that the scaling method using the entire data sets of WRCs explained 83.8% of the total spatial variability in the WRCs. The value of log(−${\widehat{\psi }}_m$) optimized using this case was similar to that using the Fitting Approaches IF and UF (Table 1). In addition, the value of $\widehat{\sigma }$ was similar to the value optimized using the Fitting Approach UF.

Consequently, the scaling method could derive reference parameters that could describe the spatial variability in the WRCs for the studied forested slope. Therefore, we conclude that scaling can be conducted with high accuracy if the reference parameters are determined appropriately.

3.3. Relationship Between Effect of Scaling, EOS, and Reference Parameters ${\widehat{\psi }}_m$ and $\widehat{\sigma }$

Figure 6 presents the relationship between reference parameters log(−ψ_m) and σ optimized for each soil sample (i.e., case of sample size 1). This panel also shows isolines of EOS and the plot showing the highest EOS (i.e., EOS for a sample size of 77) determined in the previous subsection. This analysis of Figure 6 shows data to confirm how the data of reference parameters distribute and how much EOS of each data of one WRC is, before the main analysis, the Monte Carlo approach.

In Figure 6, the lengths of both axes were set such that the standard deviations of both reference parameters were of the same length on the panel. Hence, the shape of the isolines of EOS reflects the sensitivity of EOS to both reference parameters. It is clear from the figure that the EOS showed a concentric ellipse, whose widths were narrower in the direction of log(−${\widehat{\psi }}_m$) axis than in that of the $\widehat{\sigma }$ axis. This shape indicated that the value of EOS was more sensitive to log(−${\widehat{\psi }}_m$) than $\widehat{\sigma }$.

In the following, we focus on the plots of the parameters log(−ψ_m) and σ, which correspond to the reference parameters when the sample size is one (i.e., n = 1). The plots of reference parameters were scattered widely around the highest EOS point, and it turned out that about 80% of the plots were in the area of EOS > 0.6. This proportion of plots corresponded to the confidence level of obtaining the EOS > 0.6 when we determined ${\hat{\mathsf{\psi }}}_{\mathrm{m}}$ and $\hat{\mathsf{\sigma }}$ by measuring the parameters of the WRC of one soil sample. In Figure 7, we show the cumulative relative frequency of the EOS values obtained when the sample size was 1 (i.e., n = 1). In this figure, the cumulative relative frequency represents the confidence level of obtaining an EOS value equal to or less than that indicated by the abscissa. At the 80% confidence level, an EOS of 0.6 was obtained for the sample size of 1 (i.e., n = 1). Moreover, when a lower 60% confidence level was set, the obtained EOS increased to >0.77.

3.4. Accuracy of Scaling Method for Different Sample Sizes

In this section, we present the results of a numerical experiment to examine the efficiency of the reference parameters derived from the WRCs selected using random sampling.

Figure 8 shows the plots of the reference parameters ($\log ({-\hat{\mathsf{\psi }}}_{\mathrm{m}})$ and $\hat{\mathsf{\sigma }}$) for the sample sizes n = 3, 7, 8, 20, and 30, with the isolines of EOS. For each n value, sets of reference parameters were generated using Monte Carlo simulation. The figure shows the relative frequency for each grid in the $\log ({-\hat{\mathsf{\psi }}}_{\mathrm{m}})$ and $\hat{\mathsf{\sigma }}$ ranges, which are the reference parameters and are identical for the entire dataset. The cumulative frequency distribution curves for several sample sizes are shown in Figure 7.

As previously shown in Figure 6, the parameters exhibited large variations for a sample size of 1, and the EOS obtained at a high confidence level (e.g., 80%) was low (e.g., 0.6). Figure 8 shows that the plots of the parameters were concentrated around the point showing the highest EOS with an increase in sample size. According to this change, the cumulative relative frequency increased; that is, the EOS obtained at a given confidence level increased.

In Figure 9, we show how the scaling effect increases with the sample size at a given confidence level. Using this figure, we can investigate the number of samples required to obtain a sufficient EOS. By using only eight samples, one can attain an EOS > 0.78 at a 95% confidence level. The scaling modeling method developed by Hayashi et al. (2009) [17] was applied successfully by reducing the number of reference parameters. Salemi et al. (2020) [27] proposed that the adequate sample size required to represent the spatial variability of the saturated hydraulic conductivity is approximately 20. The sample size of eight was much smaller than that estimated by Salemi et al. (2020) [27].

We explained why our number for the necessary sample size is significantly smaller than that of Salemi et al. (2020) [27]. Salemi et al. (2020) [27] did not analyze the spatial variability of reference parameters, which showed small variability, but analyzed the large variability of saturated hydraulic conductivity. Furthermore, Salemi et al. (2020) [27] and our study collected soils from different fields. Salemi et al. (2020)’s [27] field was a 40-year-old pasture, while our study’s field is a forest slope.

3.5. Efficient Sampling Strategy

As previously mentioned, many samples are required using random sampling to obtain a high EOS when a high confidence level is set. It is expected that by conducting sampling at the optimal slope segment and soil layer, rather than random sampling, the required number of WRCs can be reduced. In this section, we evaluated the most effective sampling strategy for determining the reference parameters.

For this purpose, we conducted stratified instead of uniform random sampling to select WRCs. In the stratified sampling, 77 WRCs were divided into n strata according to their depths and horizontal points (Figure 10), and one WRC dataset was chosen randomly from each stratum, resulting in n samples.

We divided the population into the following strata: In Scheme 1, the population was divided into upper and lower slope segments, each of which was subdivided into n/2 soil layers (Figure 10a). In Scheme 2, the population was divided into surface and subsurface layers, each of which was subdivided into n/2 segments in the slope direction (Figure 10b). In Scheme 3, the population was divided into n segments along the slope direction (Figure 10c). Consequently, under the condition that the same sample size was set, Scheme 1 considered the variability for the vertical direction the most and for the forest slope direction the least, whereas Scheme 3 considered the variability for the vertical direction the least and for the slope direction the most.

Figure 11 presents the cumulative relative frequency of the EOS for Schemes 1, 2, and 3 for a sample size of 8 and for random sampling for sample sizes of 8 and 10. Each of the three stratified sampling schemes performed better than random sampling for the same sample size of eight. Among the three schemes, the cumulative frequency was the highest in Scheme 3, lower in Scheme 2, and lowest in Scheme 1, respectively. The cumulative frequency for Scheme 3 was comparable to that for random sampling for a sample size of 10. Therefore, Scheme 3 could reduce the sample size by two to obtain the same EOS in comparison to the case in which no sampling strategy was introduced. Hendrayanto et al. (2000) [19] showed the spatial distribution of log(−ψ_m) observed in the entire slope. They reported that log(−ψ_m) is larger in the upper slope region than in the lower slope region. This trend supports the efficiency of Scheme 3.

3.6. Semivariogram of Reference Parameters

Scheme 3, which considered the variability for the slope direction most greatly, was most advantageous among the three schemes, whereas Scheme 1, which mostly considered the variability for the vertical direction, was least advantageous. This result suggested that the values of reference parameters had larger variability for the forest slope direction than for the vertical direction. In fact, both log(−ψ_m) and σ showed this trend in Figure 5. Note that this does not necessarily indicate that the WRC exhibits larger variability for the slope direction than for the vertical direction because the spatial variability of WRC was characterized more greatly by the scaling parameter than by the reference parameters.

Figure 12 shows the calculated semivariogram (Equation (13)). The semivariance of the log(−ψ_m) for the slope direction had a small nugget and was as large as the variance of all samples at the lag distance of 1.7 m (Figure 12a). Because the average sampling interval for the slope direction is 1.8 m, this result indicates that there were many adjacent pairs in the slope direction that had a small correlation in the log(−ψ_m) value. The same is true σ for the slope direction (Figure 12c). That is, the semivariance of the σ for the slope direction reached the variance of all samples at the lag distance < 1.8 m. For the vertical direction, both the semivariance of log(−ψ_m) and σ did not show a larger value than the nugget (Figure 12b,d).

Consequently, both reference parameters varied more in the slope direction than in the vertical direction. For both parameters, there was little correlation between the values of adjacent pairs in the slope direction. Therefore, it led to the result that Scheme 3 performed best. It is crucial to collect samples from different slope segments differentiated by geomorphological processes to determine the reference parameters for a naturally forested slope.

4. Conclusions

We estimated the number of WRCs required to determine the reference parameters (ψ_m and σ) to effectively apply the scaling technique proposed for forest soil [17]. For this purpose, we implemented random sampling of the WRCs collected from an entire forested slope by varying the sample size using the Monte Carlo method. As a result, in the figure of the relationship between log(−ψ_m) and σ, isolines of EOS were distributed concentrically. The highest EOS calculated from the whole 77 data sets of reference parameters was plotted at the center of the figure. In addition, as the sample number of WRCs required to determine reference parameters was larger, the probability of deriving the highest EOS was greater. The effect of scaling increased with increasing sample size. Its rate of increase decreased and became about 0 at a sample size of approximately 20. In total, 78% of the spatial variability of the WRCs was explained at a 95% confidence level by using the reference parameters derived from eight samples.

Stratified sampling was performed to reduce the number of WRCs required with a high EOS. Three sampling schemes were tested by considering the different spatial patterns to investigate the effective stratified sampling. The sampling scheme that considered the variability in the slope direction but not in the vertical direction was the most advantageous. The reference parameters derived from eight samples using this scheme performed the accuracy of ten samples derived from complete random sampling.

This result suggested that the spatial variability in the reference parameters is affected by geomorphological processes. Here, geomorphological processes mean an ancient landslide on the studied slope caused a thick soil layer in the downslope region consisting of debris brought from the middle slope region, and the middle and upper slope regions contained the remaining soils. We found that such geomorphological processes had a governing effect on the determination of reference parameters.

We should describe the unsaturated hydraulic conductivity. LN model functionalized unsaturated hydraulic conductivity using parameters of saturated hydraulic conductivity and two parameters (ψ_m and σ). Here, the two parameters correspond to the reference parameters used in this study. Hayashi et al. (2009) [17] insisted that reference parameters can be set to be homogeneous based on soil pore structure. Therefore, it is highly possible to assume that two parameters are homogeneous when we express unsaturated hydraulic conductivity.

In this study, we investigated datasets obtained from a steep forested slope underlain by weathered granite. Future studies should examine the applicability of the proposed scaling technique to slopes under different geomorphological, vegetative, and geological conditions.