# Daily Based Morgan–Morgan–Finney (DMMF) Model: A Spatially Distributed Conceptual Soil Erosion Model to Simulate Complex Soil Surface Configurations

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## Abstract

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## 1. Introduction

- A modified temporal scale of the model from an annual basis to daily basis. This is better suited to regions with intensive seasonal rainfall
- Inclusion of impervious surface covers (e.g., plastic mulching and artificial structures such as concrete ditches and pavements)
- Revision of the effective rainfall equation, the interflow equation, and equations relevant to flow velocity.

## 2. Model Description

#### 2.1. The DMMF Model

#### 2.2. Hydrological Phase

#### 2.2.1. Surface Runoff Process

#### 2.2.2. Interflow Process

#### 2.3. Sediment Phase

#### 2.3.1. Sediment Delivery to Surface Runoff

#### 2.3.2. Gravitational Deposition of Suspended Sediments

^{−5}$\mathrm{m}$ for clay, 0.6 × 10

^{−4}$\mathrm{m}$ for silt, and 0.2 × 10

^{−3}$\mathrm{m}$ for sand, then the settling velocities are 0.2 × 10

^{−5}$\mathrm{m}\text{}{\mathrm{s}}^{-1}$ for clay (${v}_{s.c}$), 0.2 × 10

^{−2}$\mathrm{m}\text{}{\mathrm{s}}^{-1}$ for silt (${v}_{s.z}$), and 0.2 × 10

^{−1}$\mathrm{m}\text{}{\mathrm{s}}^{-1}$ for sand (${v}_{s.s}$) [27]. The particle fall number (${N}_{f}$) of each particle size class is a function of the actual runoff velocity (v), the settling velocities of each particle size class (${v}_{s}$), the depth of runoff (d) in meters, and the length of the element (l):

#### 2.3.3. Estimation of Sediment Loss from an Element

#### 2.4. Estimation of Total Runoff and Soil Erosion for Rainfall Period

## 3. Testing the DMMF Model

#### 3.1. Sensitivity Analysis of the Model

#### 3.2. Testing the DMMF Model in the Field

## 4. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic hydrological processes within an element. The hydrological phase estimates the amount of surface runoff (Q; mm) and subsurface interflow ($I{F}_{out}$; $\mathrm{L}$) generated from an element. Assuming that the surface area of an element is A (${\mathrm{m}}^{2}$), surface water inputs of an element is the effective rainfall (${R}_{eff}$; mm) and surface water contribution from upslope elements ($\mathrm{\Sigma}\left({Q}_{in}\right)/A$; mm). Surface runoff occurs when surface water inputs exceed surface water infiltration capacity, ($S{W}_{c}$; mm) which depends on available soil pore space left for surface water infiltration and the proportion of the impervious surface area ($IMP$). The subsurface interflow occurs when the soil water budget ($SW$; mm) exceeds the soil water at field capacity ($S{W}_{fc}$; mm). In this condition, a part of the excess soil water outflows from an element as an interflow, and the surface runoff and subsurface interflow generated in an element are discharged to downslope elements.

**Figure 2.**Conceptual representation of the effective rainfall (${R}_{eff}$) on a slope element without permanent interception of rainfall (modified from Figure 1 of Choi et al. [39]). Given rainfall with a total volume of P, the amount of rainfall per unit area for both A (${\mathrm{m}}^{2}$) and ${A}^{\prime}$ (${\mathrm{m}}^{2}$) is $P/A$ and $P/{A}^{\prime}$ which is equal to R. From the trigonometric rule, A, the projected area of ${A}^{\prime}$ on the slope, is described as ${A}^{\prime}/\mathrm{cos}\left(S\right)$. Therefore, the rainfall per unit surface area of the element (i.e., the effective rainfall) should be $R\xb7\mathrm{cos}\left(S\right)$.

**Figure 3.**Conceptual representation of interflow in an element (modified from Figure 3 of Choi et al. [39]). Let’s assume that there is an element with the width of w, the length of l and slope of S. Then, given transferable soil water for interflow ($SW-S{W}_{fc}$) and saturated soil lateral hydraulic conductivity (K), the volume of interflow from the element ($I{F}_{out}$) can be represented as $K\xb7\mathrm{sin}\left(S\right)\xb7(SW-S{W}_{fc})\xb7w$, and cannot exceed the volume of the transferable soil water of the element ($(SW-S{W}_{fc})\xb7A$).

**Figure 4.**Schematic sediment phase of an element. The model estimates the amount of sediment loss from an element through three steps. In the first step, detached soil particles from an element (by raindrop (F) and runoff (H)) and sediment inputs from upslope elements ($\mathrm{\Sigma}\left(S{L}_{in}\right)/A$) are delivered to the surface water of an element. Second, some of the suspended sediments ($SS$) delivered in the runoff settle down due to gravity at the deposition rate of the suspended sediments in the runoff ($DEP$). Third, the model estimates the amount of sediment loss from an element by comparing the transport capacity of the runoff ($TC$) and sediments available for transport (G), which are the remaining suspended sediments after gravitational deposition process. If $TC$ is larger than G, all the remaining sediments in the water (i.e., G) are washed away from an element. Otherwise, the amount of sediments equal to $TC$ is carried out by the surface runoff to downslope elements.

**Figure 5.**Sobol’ total indices of model input parameters for a single element. The bars indicate the Sobol’ total indices and the error bars indicate the 95% confidence intervals of the indices from bootstrapping.

**Figure 6.**Sobol’ total indices for runoff (Q) and sediment loss ($SL$) of the two field sites. Bars indicate the Sobol’ total indices and the error bars indicate the 95% confidence intervals of the indices from bootstrapping. We checked the sensitivity of the model to the parameters with high uncertainty due to absence of field data, such as parameters related to soil detachability (i.e., $D{K}_{c}$, $D{K}_{z}$, $D{K}_{s}$, $D{R}_{c}$, $D{R}_{z}$, and $D{R}_{s}$), soil hydraulic parameters (i.e., K, ${\theta}_{sat}$, and ${\theta}_{fc}$), vegetation structural parameters (i.e., $GC$, D, $NV$), the permanent interception ($PI$), and the rill depth (d). K, ${\theta}_{sat}$ and ${\theta}_{fc}$ showed relatively high impacts on the runoff (Q) and the sediment loss ($SL$). The sediment loss ($SL$) also showed high sensitivity to $D{R}_{c}$, $D{R}_{z}$, $D{R}_{s}$ and d.

**Figure 7.**Comparison between simulated and observed runoff (Q) and sediment loss ($SL$) for field 1 and field 2. We tested the model performance for both fields with optimized parameters (Table 3). Model performance was evaluated using the Nash-Sutcliffe efficiency coefficient (NSE), percent bias (PBIAS), and RMSE-observation standard deviation ratio (RSR) with the observed data from Arnhold et al. [37]. To make all overlapping points with values close to zero visible, we slightly jitterred the points.

Parameter | Description | Unit | Range |
---|---|---|---|

R | Daily rainfall | [mm] | 1–1825 ^{(a)} |

$RI$ | Mean rainfall intensity of a day | [mm ${\mathrm{h}}^{-1}$] | 15.0–305.0 ^{(a)} |

$ET$ | Daily evapotranspiration | [mm] | 0.0–15.0 ^{(b)} |

S | Slope angle | [$\mathrm{rad}$] | 0.0–1.5 ^{(c)} |

$res$ | Grid size of a raster map for the width (w) and the length (l)of an element that are equal to $res$ and $res/\mathrm{cos}\left(S\right)$ | [$\mathrm{m}$] | 0.25–100 ^{(d)} |

${P}_{c}$ | Proportion of clay of the surface soil | [proportion] | 0–1 |

${P}_{z}$ | Proportion of silt of the surface soil | [proportion] | 0–1 |

${P}_{s}$ | Proportion of sand of the surface soil | [proportion] | 0–1 |

$SD$ | Soil depth | [$\mathrm{m}$] | 0.3–68.0 ^{(e)} |

${\theta}_{init}$ | Initial soil water content of entire soil profile | [$\mathrm{vol}/\mathrm{vol}$] | 0.00–${\theta}_{sat}$ ^{(f)} |

${\theta}_{sat}$ | Saturated water content of entire soil profile | [$\mathrm{vol}/\mathrm{vol}$] | 0.31–0.56 ^{(f)} |

${\theta}_{fc}$ | Soil water content at field capacity of entire soil profiles | [$\mathrm{vol}/\mathrm{vol}$] | 0.10–${\theta}_{sat}$ ^{(f)} |

K | Saturated soil lateral hydraulic conductivity | [$\mathrm{m}\text{}{\mathrm{d}}^{-1}$] | 1–230 ^{(g)} |

$D{K}_{c}$ | Detachability of clay particles by rainfall | [$\mathrm{g}\text{}{\mathrm{J}}^{-1}$] | 0.10–1.50 ^{(h)} |

$D{K}_{z}$ | Detachability of silt particles by rainfall | [$\mathrm{g}\text{}{\mathrm{J}}^{-1}$] | 0.50–5.15 ^{(h)} |

$D{K}_{s}$ | Detachability of sand particles by rainfall | [$\mathrm{g}\text{}{\mathrm{J}}^{-1}$] | 0.15–4.15 ^{(h)} |

$D{R}_{c}$ | Detachability of clay particles by surface runoff | [$\mathrm{g}\text{}{\mathrm{mm}}^{-1}$] | 0.020–2.0 ^{(h)} |

$D{R}_{z}$ | Detachability of silt particles by surface runoff | [$\mathrm{g}\text{}{\mathrm{mm}}^{-1}$] | 0.016–1.6 ^{(h)} |

$D{R}_{s}$ | Detachability of sand particles by surface runoff | [$\mathrm{g}\text{}{\mathrm{mm}}^{-1}$] | 0.015–1.5 ^{(h)} |

$PI$ | Area proportion of the permanent interception of rainfall | [proportion] | 0–1 |

$IMP$ | Area proportion of the impervious ground cover | [proportion] | 0–1 |

$GC$ | Area proportion of the ground cover of the soil surfaceprotected by vegetation or crop cover on the ground | [proportion] | 0–1 |

$CC$ | Area proportion of the canopy cover of the soil surfaceprotected by vegetation or crop canopy | [proportion] | 0–1 |

$PH$ | Average height of vegetation or crop cover of an elementwhere leaf drainage starts to fall | [$\mathrm{m}$] | 0–30 ^{(h)} |

D | Average diameter of individual plant elements at the surface | [$\mathrm{m}$] | 0.00001–3.0 ^{(h)} |

$NV$ | Number of individual plant elements per unit area | [$\mathrm{number}/{\mathrm{m}}^{2}$] | 0.00001–2000 ^{(h)} |

d | Typical flow depth of surface runoff in an element | [$\mathrm{m}$] | 0.005–3 ^{(h)} |

n | Manning’s roughness coefficient of the soil surface | [$\mathrm{s}\text{}{\mathrm{m}}^{-1/3}$] | 0.01–0.05 ^{(i)} |

^{(a)}is based on WMO [56];

^{(b)}is based on Senay et al. [57], Jia et al. [58];

^{(c)}represents the range of slope from a flat surface to a vertical cliff;

^{(d)}is based on Lilhare et al. [17], Arnhold et al. [37], Pandey et al. [59];

^{(e)}is based on the range of rooting depth from Canadell et al. [60];

^{(f)}is based on Saxton et al. [61];

^{(g)}is based on the hydraulic conductivity of semi-pervious soils from Irmay [62];

^{(h)}is based on Morgan and Duzant [27];

^{(i)}is based on Manning’s n of bare soil in Table 3.6 from Morgan [3].

Field | ${\mathit{\theta}}_{\mathit{sat}}$ | ${\mathit{\theta}}_{\mathit{fc}}$ | K | $\mathit{PI}$ | $\mathit{GC}$ | $\mathit{NV}$ | D | ${\mathit{DK}}_{\mathit{c}}$ | ${\mathit{DK}}_{\mathit{z}}$ | ${\mathit{DK}}_{\mathit{s}}$ | ${\mathit{DR}}_{\mathit{c}}$ | ${\mathit{DR}}_{\mathit{z}}$ | ${\mathit{DR}}_{\mathit{s}}$ | d | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Field 1 | Sup. | 0.454 | 0.351 | 17.9 | 0.144 | 0.48 | 5.4 | 0.12 | 1.50 | 5.15 | 4.15 | 2.0 | 1.6 | 1.5 | 0.010 |

Inf. | 0.351 | 0.345 | 0.29 | 0.096 | 0.32 | 3.6 | 0.08 | 0 | 0 | 0 | 0 | 0 | 0 | 0.005 | |

Field 2 | Sup. | 0.494 | 0.435 | 5.22 | 0.144 | 0.48 | 5.4 | 0.12 | 1.50 | 5.15 | 4.15 | 2.0 | 1.6 | 1.5 | 0.010 |

Inf. | 0.435 | 0.407 | 0.15 | 0.096 | 0.32 | 3.6 | 0.08 | 0 | 0 | 0 | 0 | 0 | 0 | 0.005 |

K | ${\mathit{\theta}}_{\mathit{sat}}$ | ${\mathit{\theta}}_{\mathit{fc}}$ | ${\mathit{DR}}_{\mathit{c}}$ | ${\mathit{DR}}_{\mathit{z}}$ | ${\mathit{DR}}_{\mathit{s}}$ | d | |
---|---|---|---|---|---|---|---|

Field 1 | 0.500 | 0.362 | 0.345 | 0.015 | 0.012 | 0.011 | 0.010 |

Field 2 | 0.284 | 0.453 | 0.435 | 0.007 | 0.005 | 0.005 | 0.005 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Choi, K.; Arnhold, S.; Huwe, B.; Reineking, B.
Daily Based Morgan–Morgan–Finney (DMMF) Model: A Spatially Distributed Conceptual Soil Erosion Model to Simulate Complex Soil Surface Configurations. *Water* **2017**, *9*, 278.
https://doi.org/10.3390/w9040278

**AMA Style**

Choi K, Arnhold S, Huwe B, Reineking B.
Daily Based Morgan–Morgan–Finney (DMMF) Model: A Spatially Distributed Conceptual Soil Erosion Model to Simulate Complex Soil Surface Configurations. *Water*. 2017; 9(4):278.
https://doi.org/10.3390/w9040278

**Chicago/Turabian Style**

Choi, Kwanghun, Sebastian Arnhold, Bernd Huwe, and Björn Reineking.
2017. "Daily Based Morgan–Morgan–Finney (DMMF) Model: A Spatially Distributed Conceptual Soil Erosion Model to Simulate Complex Soil Surface Configurations" *Water* 9, no. 4: 278.
https://doi.org/10.3390/w9040278