# Identification of the Representative Point for Soil Moisture Storage Using a Precipitation History Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}and 4000 m

^{2}, and the average slopes were 19° and 28°, respectively. The soil texture was commonly a mixture of sandy loam and loamy sand with sand (43–65%), silt (29–46%), and clay (6–11%) on hillslope B and sandy loam and loamy sand with sand (57–76%), silt (22–38%), and clay (5–10%) on hillslope S, respectively, based on soil taxonomy (USDA). Both hillslopes exhibited abundant macropores in the topsoil on visual inspection. The mean soil porosities were 49.6% and 48% for hillslopes B and S, respectively. The soil thickness of hillslope B was 40–100 cm and that of hillslope S was 25–90 cm [15,16]. The soil layer on hillslope B was well distributed between the upslope and downslope, whereas the spatial distribution of soil thickness was relatively uneven on hillslope S. A canopy mixture of Carpinus sp. and shrubby Quercus sp. was the primary land cover on hillslope B, with no notable spatial distribution of vegetation communities. Hillslope S was covered by a mixture of Polemoniales, shrubby Quercus sp., and a coniferous canopy of Pinus densiflora. A flux tower is located 300 m downstream of hillslope B, which has been operating since 2005, collecting fluxes of H

_{2}O and CO

_{2}through the eddy covariance technique [17]. Evapotranspiration data were also obtained using the eddy covariance method at a flux tower 50 m from hillslope S.

#### 2.2. Acquiring Hydrologic Data on the Hillslope

#### 2.3. Mathematical Development for Soil Water Storage

#### 2.4. Prediction of Soil Moisture Using Rainfall History

#### 2.5. Stochastic Model for the Difference between PHI and SWS

## 3. Results

#### 3.1. Relationship between SWS and PHI

^{2}and root mean square error (RMSE) in the validation dataset is lower than that in the calibration dataset for both hillslopes (see Table 1), indicating high sensitivity and uncertainty in K for the difference in rainfall features between the two datasets.

#### 3.2. Stochastic Models for the Difference between PH and SWS

#### 3.3. Representative Points Based on the Stochastic Process of SWS and Temporal Stability

^{2}values than the conventional temporal stability analysis for the linear relationships between average SWS and the selected point at selected points.

^{2}and RMSE) for calibration and validation between the spatially averaged SWS and the SWS for the representative point at three depths for the two hillslopes. The representative points based on the stochastic processes of SWS exhibited higher R

^{2}and lower RMSE than those based on temporal stability in both the training dataset (calibration) and the testing set (validation) (Table 4).

## 4. Discussion

#### 4.1. Predictability of SWS

^{2}and RMSE. The PHI model exhibited approximately similar predictability (0.34 < R

^{2}< 0.63) in both calibration and validation datasets. The linear model (Equation (4)) had an identical R

^{2}to the PHI model; however, the RMSEs of the linear model were substantially lower, indicating the fitting capability of the linear model for the bias of the PHI model. The exponential or empirical models shown in Equations (5) and (6) exhibited similar performances to the linear model (Table 5). The SP model for the average SWS showed distinctly higher R

^{2}values (between 0.87 and 0.96) and substantially smaller RMSEs (1.55–10.61) than other existing models, as presented in Table 5. The prediction using the identified representative points at C3 (hillslope B) and C6 (hillslope S) based on SP models also exhibited comparable performance both in R

^{2}values (0.78–0.92) and RMSE values (2.08–12.23).

#### 4.2. Hydrological Interpretation of Modeling Results

_{t}in Equations (9) and (10) reveals that evapotranspiration played an important role in the prediction of SWS. The minor difference in estimated parameters between soil moisture measurement and flux tower measurement can be explained by the data acquisition conditions and hydrological processes in the soil layer. The definition of ET

_{t}in Equations (1)–(3) may further consider the exfiltration process in the soil layer, which naturally has a longer memory impact than that of the flux tower.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Locations of the Bongsunsa catchment, Sulmachun catchment (

**a**), and study areas (hillslope B (

**b**) and hillslope S (

**c**)) with the surface digital elevation model and measurement systems for soil moisture and precipitation.

**Figure 2.**Index of temporal stability (ITS) for points at three depths (10 cm, 30 cm, 60 cm) and average ITS of each measurement point for (

**a**) hillslope B and (

**b**) hillslope S.

**Figure 3.**Relationship between the average SWS and SWS for the representative point based on the stochastic structure of (

**a**) point C3 at hillslope B and (

**c**) point C6 at hillslope S and the representative point based on the temporal stability of (

**b**) point B3 at hillslope B and (

**d**) point B6 at hillslope S. * and ** in equations indicate significant level p < 0.05 and p < 0.01, respectively.

**Table 1.**Calibrated K coefficients and predictabilities in the PHI model for soil water storage (SWS) at all depths (20 cm, 45 cm, and 80 cm) for the two hillslopes.

Hillslope | B | S | |||||
---|---|---|---|---|---|---|---|

soil depth for storage | 20 cm | 45 cm | 80 cm | 20 cm | 45 cm | 80 cm | |

coefficient K | 0.83 | 0.85 | 0.87 | 0.93 | 0.94 | 0.95 | |

calibration data | ${\mathrm{R}}^{2}$ | 0.66 | 0.56 | 0.54 | 0.42 | 0.42 | 0.44 |

$\mathrm{RMSE}$ | 41.07 | 45.37 | 51.2 | 35.67 | 35.85 | 73.99 | |

validation data | ${\mathrm{R}}^{2}$ | 0.57 | 0.54 | 0.57 | 0.34 | 0.34 | 0.35 |

$\mathrm{RMSE}$ | 90.36 | 99.85 | 113.08 | 23.80 | 40.18 | 92.19 |

^{2}: coefficient of determination; RMSE: root mean square error.

**Table 2.**Estimated parameters and SP models for SWS at depths of 20 cm, 45 cm, and 80 cm for all monitoring points and residual tests for hillslope B. Black dots (●) represent the residual as white noise through a

**χ**test; the rings (○) denote stochastic structures in the residuals.

^{2}20 cm | Avg | A1 | A2 | A3 | A4 | C1 | C2 | C3 | C4 |
---|---|---|---|---|---|---|---|---|---|

AR(1) | 0.82 | 0.88 | 0.94 | 0.66 | 0.48 | 1.08 | 0.94 | 0.57 | 0.69 |

AR(2) | −0.14 | ||||||||

AR(3) | 0.16 | 0.15 | 0.26 | 0.42 | |||||

AR(4) | −0.24 | −0.34 | −0.27 | ||||||

AR(5) | 0.18 | 0.23 | 0.29 | ||||||

${\mathit{\chi}}^{2}$ | ● | ● | ● | ○ | ○ | ● | ● | ● | ○ |

45 cm | Avg | A1 | A2 | A3 | A4 | C1 | C2 | C3 | C4 |

AR(1) | 0.83 | 0.9 | 0.95 | 0.71 | 0.47 | 0.96 | 1.10 | 0.65 | 0.70 |

AR(2) | −0.15 | ||||||||

AR(3) | 0.14 | 0.13 | 0.23 | 0.36 | |||||

AR(4) | −0.18 | −0.35 | −0.30 | ||||||

AR(5) | 0.15 | 0.11 | 0.25 | 0.27 | |||||

${\mathit{\chi}}^{2}$ | ● | ● | ● | ○ | ○ | ● | ● | ● | ○ |

80 cm | Avg | A1 | A2 | A3 | A4 | C1 | C2 | C3 | C4 |

AR(1) | 0.82 | 0.95 | 0.57 | 0.97 | 1.12 | 0.70 | 0.72 | ||

AR(2) | −0.17 | ||||||||

AR(3) | 0.18 | 0.20 | 0.35 | ||||||

AR(4) | −0.19 | −0.39 | −0.27 | ||||||

AR(5) | 0.15 | 0.12 | 0.28 | 0.25 | |||||

${\mathit{\chi}}^{2}$ | ● | ● | ○ | ● | ● | ● | ○ | ||

20 cm | B1 | B2 | B3 | B4 | B5 | D1 | D2 | D3 | D4 |

AR(1) | 0.93 | 0.93 | 0.88 | 0.67 | 0.74 | 0.91 | 0.94 | 0.76 | 0.62 |

AR(2) | −0.22 | ||||||||

AR(3) | 0.18 | 0.12 | 0.21 | ||||||

AR(4) | −0.26 | −0.19 | −0.29 | ||||||

AR(5) | 0.25 | 0.31 | 0.25 | ||||||

${\mathit{\chi}}^{2}$ | ● | ● | ● | ● | ○ | ● | ● | ○ | ● |

45 cm | B1 | B2 | B3 | B4 | B5 | D1 | D2 | D3 | D4 |

AR(1) | 0.96 | 0.96 | 0.85 | 0.65 | 0.76 | 0.94 | 0.97 | 0.82 | 0.67 |

AR(2) | −0.21 | ||||||||

AR(3) | 0.21 | 0.14 | 0.24 | ||||||

AR(4) | −0.29 | −0.21 | −0.23 | ||||||

AR(5) | 0.24 | 0.32 | 0.25 | ||||||

${\mathit{\chi}}^{2}$ | ● | ● | ● | ● | ○ | ● | ● | ○ | ○ |

80 cm | B1 | B2 | B3 | B4 | B5 | D1 | D2 | D3 | D4 |

AR(1) | 1.01 | 0.90 | 0.62 | 0.79 | 0.95 | 0.93 | |||

AR(2) | |||||||||

AR(3) | 0.15 | ||||||||

AR(4) | −0.18 | −0.25 | |||||||

AR(5) | 0.13 | 0.13 | 0.21 | ||||||

${\mathit{\chi}}^{2}$ | ● | ● | ● | ○ | ● | ● |

Hillslope | Datasets | Mean | Median | Standard Deviation | Max. | Min. | |
---|---|---|---|---|---|---|---|

B | 20 cm SWS | ${e}_{t}$ Equation (9) | 27.74 | 16.19 | 38.25 | 245.14 | −0.01 |

Equation (10) | 28.82 | 33.05 | 18.20 | 58.25 | 0.55 | ||

45 cm SWS | ${e}_{t}$ Equation (9) | 31.82 | 24.45 | 35.16 | 229.01 | −1.16 | |

Equation (10) | 32.65 | 37.22 | 20.47 | 65.17 | 0.55 | ||

80 cm SWS | ${e}_{t}$ Equation (9) | 36.31 | 30.93 | 33.15 | 214.02 | −1.87 | |

Equation (10) | 37.66 | 43.82 | 23.41 | 74.08 | 0.55 | ||

S | 20 cm SWS | ${e}_{t}$ Equation (9) | 41.83 | 31.47 | 33.45 | 186.63 | −2.35 |

Equation (10) | 44.97 | 52.08 | 21.27 | 78.32 | 0.69 | ||

45 cm SWS | ${e}_{t}$ Equation (9) | 51.61 | 44.72 | 30.90 | 174.85 | −5.58 | |

Equation (10) | 52.27 | 60.16 | 24.19 | 89.59 | 0.69 | ||

80 cm SWS | ${e}_{t}$ Equation (9) | 65.66 | 70.32 | 30.34 | 156.79 | −10.59 | |

Equation (10) | 62.43 | 71.02 | 28.75 | 104.86 | 0.69 |

**Table 4.**Representative points determined by the similarity between the stochastic model of SWS and temporal stability and their regression relationship to average SWS with R

^{2}and RMSE.

Hillslope | Dataset | Point | Calibration | Validation | ||
---|---|---|---|---|---|---|

${\mathit{R}}^{2}$ | $\mathbf{RMSE}$ | ${\mathit{R}}^{2}$ | $\mathbf{RMSE}$ | |||

B | 20 cm SWS | C3 | 0.83 | 2.99 | 0.86 | 4.84 |

B3 | 0.74 | 3.65 | 0.83 | 9.06 | ||

45 cm SWS | C3 | 0.82 | 6.03 | 0.89 | 8.77 | |

B3 | 0.77 | 6.86 | 0.87 | 12.10 | ||

80 cm SWS | C3 | 0.83 | 8.95 | 0.93 | 10.50 | |

B3 | 0.81 | 9.42 | 0.89 | 18.40 | ||

S | 20 cm SWS | C6 | 0.95 | 1.64 | 0.90 | 2.59 |

B6 | 0.94 | 1.74 | 0.90 | 4.05 | ||

45 cm SWS | C6 | 0.96 | 3.54 | 0.96 | 5.06 | |

B6 | 0.93 | 4.43 | 0.92 | 6.33 | ||

80 cm SWS | C6 | 0.96 | 6.28 | 0.98 | 8.00 | |

B6 | 0.94 | 7.28 | 0.92 | 8.95 |

**Table 5.**Predictabilities (R

^{2}and RMSE) of SWS using existing regression models (Lin.: linear; Exp.: exponential; [20] empirical) and the stochastic precipitation (SP) model. SP(C3) is the SP model from point C3.

Hillslope | Dataset | Model | PHI(K) | Lin. | Exp. | Empirical[20] | SP | SP(C3) | |

Input Variables | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$${\overline{\mathit{\theta}}}_{\mathit{t}\mathbf{-}\mathit{k}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$${\mathit{\theta}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | |||

B | 20 cm SWS | calibration 2009 | ${\mathrm{R}}^{2}$ | 0.66 | 0.66 | 0.68 | 0.67 | 0.93 | 0.83 |

$\mathrm{RMSE}$ | 44.55 | 4.16 | 4.10 | 4.12 | 1.85 | 3.03 | |||

validation 2011 | ${\mathrm{R}}^{2}$ | 0.57 | 0.57 | 0.71 | 0.73 | 0.93 | 0.78 | ||

$\mathrm{RMSE}$ | 83.48 | 8.37 | 6.03 | 5.81 | 3.05 | 5.50 | |||

45 cm SWS | calibration 2009 | ${\mathrm{R}}^{2}$ | 0.63 | 0.63 | 0.63 | 0.63 | 0.94 | 0.80 | |

$\mathrm{RMSE}$ | 89.44 | 8.68 | 8.60 | 8.58 | 3.35 | 6.36 | |||

validation 2011 | ${\mathrm{R}}^{2}$ | 0.57 | 0.57 | 0.68 | 0.71 | 0.94 | 0.83 | ||

$\mathrm{RMSE}$ | 101.46 | 16.20 | 12.48 | 11.89 | 5.42 | 9.75 | |||

80 cm SWS | calibration 2009 | ${\mathrm{R}}^{2}$ | 0.61 | 0.61 | 0.62 | 0.62 | 0.95 | 0.80 | |

$\mathrm{RMSE}$ | 170.67 | 13.45 | 13.36 | 13.26 | 4.78 | 9.67 | |||

validation 2011 | ${\mathrm{R}}^{2}$ | 0.57 | 0.57 | 0.67 | 0.71 | 0.94 | 0.88 | ||

$\mathrm{RMSE}$ | 168.22 | 23.91 | 18.67 | 17.55 | 7.83 | 12.22 | |||

Hillslope | Dataset | Model | PH(K) | Lin. | Exp. | Empirical[20] | SP | SP(C6) | |

Input Variables | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$${\overline{\mathit{\theta}}}_{\mathit{t}\mathbf{-}\mathit{k}}$ | $\mathit{A}\mathit{M}{\mathit{O}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$${\mathit{\theta}}_{\mathbf{1}\mathbf{,}\mathbf{\cdots}\mathbf{,}\mathit{t}}$ | |||

S | 20 cm SWS | calibration 2015 | ${\mathrm{R}}^{2}$ | 0.41 | 0.41 | 0.47 | 0.50 | 0.95 | 0.90 |

$\mathrm{RMSE}$ | 35.66 | 5.57 | 5.28 | 5.14 | 1.55 | 2.08 | |||

validation 2016 | ${\mathrm{R}}^{2}$ | 0.33 | 0.33 | 0.37 | 0.40 | 0.87 | 0.81 | ||

$\mathrm{RMSE}$ | 23.79 | 6.74 | 6.69 | 6.59 | 2.53 | 3.29 | |||

45 cm SWS | calibration 2015 | ${\mathrm{R}}^{2}$ | 0.42 | 0.42 | 0.47 | 0.50 | 0.96 | 0.91 | |

$\mathrm{RMSE}$ | 35.85 | 12.73 | 12.13 | 11.81 | 2.72 | 4.46 | |||

validation 2016 | ${\mathrm{R}}^{2}$ | 0.33 | 0.33 | 0.37 | 0.39 | 0.88 | 0.86 | ||

$\mathrm{RMSE}$ | 40.17 | 16.02 | 15.90 | 15.65 | 5.52 | 7.17 | |||

80 cm SWS | calibration 2015 | ${\mathrm{R}}^{2}$ | 0.43 | 0.43 | 0.46 | 0.48 | 0.96 | 0.92 | |

$\mathrm{RMSE}$ | 73.38 | 23.17 | 22.59 | 22.07 | 4.50 | 7.79 | |||

validation 2016 | ${\mathrm{R}}^{2}$ | 0.34 | 0.34 | 0.36 | 0.38 | 0.87 | 0.87 | ||

$\mathrm{RMSE}$ | 92.12 | 30.08 | 29.88 | 29.41 | 10.61 | 12.23 |

^{2}: coefficient of determination; RMSE: root mean square error.

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**MDPI and ACS Style**

Kim, S.; Lee, E.
Identification of the Representative Point for Soil Moisture Storage Using a Precipitation History Model. *Water* **2023**, *15*, 3921.
https://doi.org/10.3390/w15223921

**AMA Style**

Kim S, Lee E.
Identification of the Representative Point for Soil Moisture Storage Using a Precipitation History Model. *Water*. 2023; 15(22):3921.
https://doi.org/10.3390/w15223921

**Chicago/Turabian Style**

Kim, Sanghyun, and Eunhyung Lee.
2023. "Identification of the Representative Point for Soil Moisture Storage Using a Precipitation History Model" *Water* 15, no. 22: 3921.
https://doi.org/10.3390/w15223921