# Revised SEDD (RSEDD) Model for Sediment Delivery Processes at the Basin Scale

^{*}

## Abstract

**:**

_{i}). The use of probability to model SDR

_{i}in SEDD led us to examine the model and check for its statistical validity. As a result, we found that the SEDD model had several false assertions and needs to be revised to correct for the discrepancies with the statistical properties of the exponential distributions. The results of our study are presented here. We propose an alternative model, the Revised SEDD (RSEDD) model, to better estimate SDR

_{i}. We also show how to calibrate the model parameters and examine an example watershed to see if the travel time of sediments follows an exponential distribution. Finally, we reviewed studies citing the SEDD model to explore if they would be impacted by switching to the proposed RSEDD model.

## 1. Introduction

#### 1.1. SEDD Model

_{i}, of each morphological area is a measurement of the probability that the eroded particles arrive from the considered area into the nearest stream reach” [9]. Furthermore, the authors defined the travel time as “the time that particles eroded from the source area and transported through the hillslope conveyance system take to arrive at the channel network” [9]. Assuming that F

_{i}is the cumulative distribution function (CDF) of the travel time t

_{p,i}, the authors assert that the relationship between lnF

_{i}and t

_{p,i}is linear. Then, they support the assertion with data from seven Sicilian basins. As a result, the following exponential function was used to describe the relationship between the SDR

_{i}and the travel time:

_{i}is the SDR of morphological unit i, β is a constant for a given basin (1/m), and t

_{p,i}is the travel time of morphological unit i (m) and defined as:

_{i}can be represented as:

_{p}= the number of morphological units localized along the hydraulic path j, and λ

_{i,j}and s

_{i,j}= the length (m) and slope (m/m) of each morphological unit i localized along the hydraulic path j.

_{w}(SDR for the entire basin) to estimate the movement of sediments and their impact on particular watersheds [10,11,12,13,14]. However, our examination of the SEDD model led us to believe that the model might have been inadequately formulated to represent the probability concept declared. We will show why we think so and present our revised version of the model in the following sections.

#### 1.2. Incorrect Assertions of SEDD

- (a)
- “the Sediment Delivery Ratio, SDR
_{i}, of each morphological area is a measurement of the probability that the eroded particles arrive from the considered area into the nearest stream reach” [9]; - (b)
- the SDR
_{i}equation is an exponential distribution (exponential probability distribution); - (c)
- (d)
- “the probability that the eroded particles arrive from the morphological unit into the nearest stream reach is assumed proportional to the probability of non-exceedance of the travel time, t
_{p,i}” [9]; - (e)
- the β coefficient can be lumped together with “the effects due to roughness and runoff along the hydraulic path” [8];
- (f)
- the F
_{i}is a CDF of the travel time represented by Equation (2).

## 2. Analysis

#### 2.1. Properties of Exponential Distributions

^{x}is the exponential function and C is a constant.

_{i}in SEDD). The PDF and CDF of the exponential distribution of typical values of $\lambda $ are shown in Figure 1a,b, respectively.

#### 2.2. Examination of SEDD Assertions

_{i}.

_{i}as the probability that the eroded particles arrive from the source area into the nearest river channel. This seems to imply that they consider SDR

_{i}to be the PDF of travel time, t

_{p,i}. On the other hand, they also wrote that the probability as mentioned above is “proportional to the probability of non-exceedance of the travel time”. This statement seems to suggest that they consider SDR

_{i}to be the CDF of travel time instead. Although the statements might be contradictory to each other, it would not matter because neither is correct. To formulate the relationship between SDR

_{i}and t

_{p,i}, the authors of SEDD decided to use an exponential function as shown in Equation (1) and repeated here as Equation (9):

_{i}and t

_{p,i}to explain the observed linear data from the seven Sicilian basins. However, there are a few critical problems. First, the total area under a PDF has to be equal to one:

_{i}and t

_{p,i}such as that in Equation (4) does not exist. The only condition that Equation (4) is valid occurs when Equation (9) is a CDF. Since we have already shown that Equation (9) is not a CDF, we can conclude that assertion (c) and Equation (4) are incorrect.

_{p,i}of each morphological unit increases with the increase of the length of the hydraulic path (${l}_{p,i}$) and with the decrease of the square root of the slope of the hydraulic path (${s}_{p,i}$):

_{p,i}and the product of ${l}_{p,i}$ and $1/\sqrt{{s}_{p,i}}$. The SEDD model lumps together the constant and the β coefficient, which changes the coefficient β from its original meaning. Therefore, we think assertion (e) is not appropriate. We will correct these problems by presenting the Revised SEDD (RSEDD) model in the next section.

#### 2.3. RSEDD Model

_{p,i}is an exponential random variable. The CDF of Equation (16) is the integration of Equation (16):

_{i}and t

_{p,i}is not linear as was suggested by the SEDD model. On the contrary, Figure 2a reveals that the natural logarithm of the exponential PDF is linear. We will use this linear property to solve for the model parameters of RSEDD later.

_{p,i}of each morphological unit increases with the increase of the ratio of the length of the hydraulic path (${l}_{p,i}$) to the square root of the slope of the hydraulic path (${s}_{p,i}$). Therefore,

_{p,i}and ${l}_{p,i}/\sqrt{{s}_{p,i}}$ (representing “the effects due to roughness and runoff along the hydraulic path”) and the β coefficient. However, this would change the coefficient β from its original meaning. We will introduce a new constant k (dimensionless) and re-write Equation (2) as follows:

_{p,i}is the pseudo travel time (m). Therefore,

_{p}= the number of morphological units localized along the hydraulic path j, and λ

_{i,j}and s

_{i,j}= the length (m) and slope (m/m) of each morphological unit i localized along the hydraulic path j.

_{i}is the “probability density” of the eroded particles arriving from the considered area into the nearest stream reach. To solve for model parameters β and k, take the natural logarithm of Equation (21):

_{i}) against d

_{p,i}, we can determine β from the intercept and k from the slope of the linear plot (similar to Figure 2a). In other words, given SDR

_{i}and pseudo travel time d

_{p,i}, we can determine β and k and use RSEDD to model sediment delivery at the basin scale.

## 3. Example Watershed

_{0}: the travel times follow an exponential distribution

_{a}: the travel times do not follow an exponential distribution

_{p,i}in Table 1 from the smallest to the largest, we can calculate the cumulative distribution of our sample watershed and the corresponding cumulative distribution of the exponential distribution as shown in Table 2. For a confidence coefficient (1–α) of 0.95, we obtained the sample statistic of 0.213 (D

_{n}). Since D

_{n}is not greater than the critical value of 0.375 (D

_{12,0.05}), we cannot reject the null hypothesis (H

_{0}) that these data come from an exponential distribution. However, we cannot accept the null hypothesis (H

_{0}) either because we only know a Type I error (probability equal to α = 0.05) and do not know the probability of making a Type II error [21]. The hypothesis testing on this example watershed is not conclusive.

## 4. Discussion

## 5. Summary and Conclusions

_{i}and sediment yield in future watershed research.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The (

**a**) probability distribution function (PDF) and the (

**b**) cumulative distribution function (CDF) of the exponential distribution of typical values of $\lambda $.

**Figure 2.**The natural logarithm of Figure 1a,b (Y-axis values only). This shows that the relationship between ln(CDF) and t

_{p,i}is not linear (

**b**), but the relationship between ln(PDF) and t

_{p,i}is linear (

**a**).

**Figure 3.**Illustration of the morphological unit i and the path traveled by the sediments from the morphological unit i to the nearest river in a basin.

**Figure 4.**An example watershed (re-drawn from [20]).

Morphological Unit | Hydraulic Path | ${\mathit{\lambda}}_{\mathit{i},\mathit{j}}$ | ${\mathit{s}}_{\mathit{i},\mathit{j}}$ | $\frac{{\mathit{\lambda}}_{\mathit{i},\mathit{j}}}{\sqrt{{\mathit{s}}_{\mathit{i},\mathit{j}}}}$ | d_{p,i} |
---|---|---|---|---|---|

1 | 1 | 1.0 | 0.3 | 1.83 | 1.83 |

2 | 2 | 2.7 | 0.3 | 4.93 | 4.93 |

3 | 3 | 2.6 | 0.3 | 4.75 | 4.75 |

4 | 4 | 1.6 | 0.3 | 2.92 | 2.92 |

5 | 5 | 3.9 | 0.3 | 7.12 | 7.12 |

6 | 6–7 | 1.6 | 0.3 | 2.92 | 6.02 |

7 | 7 | 1.7 | 0.3 | 3.10 | 3.10 |

8 | 8–9 | 0.9 | 0.3 | 1.64 | 2.92 |

9 | 9 | 0.7 | 0.3 | 1.28 | 1.28 |

10 | 10 | 2.6 | 0.3 | 4.75 | 4.75 |

11 | 11 | 2.1 | 0.3 | 3.83 | 3.83 |

12 | 12 | 1.3 | 0.3 | 2.37 | 2.37 |

x | Frequency | Cumulative | Cumulative (%) | Corresponding Exponential CDF (%) | Difference | |
---|---|---|---|---|---|---|

1.28 | 1 | 1 | 0.083 | 0.284 | 0.201 | |

1.83 | 1 | 2 | 0.167 | 0.380 | 0.213 | |

2.37 | 1 | 3 | 0.250 | 0.463 | 0.213 | |

2.92 | 1 | 4 | 0.333 | 0.535 | 0.201 | |

2.92 | 1 | 5 | 0.417 | 0.535 | 0.118 | |

3.10 | 1 | 6 | 0.500 | 0.556 | 0.056 | |

3.83 | 1 | 7 | 0.583 | 0.634 | 0.050 | |

4.75 | 1 | 8 | 0.667 | 0.711 | 0.045 | |

4.75 | 1 | 9 | 0.750 | 0.711 | 0.039 | |

4.93 | 1 | 10 | 0.833 | 0.725 | 0.108 | |

6.02 | 1 | 11 | 0.917 | 0.794 | 0.123 | |

7.12 | 1 | 12 | 1.000 | 0.845 | 0.155 | |

Total | 12 | |||||

Mean | 3.82 | D_{n} | 0.213 | |||

λ | 0.262 | D_{12,0.05} | 0.375 |

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**MDPI and ACS Style**

Chen, W.; Thomas, K.
Revised SEDD (RSEDD) Model for Sediment Delivery Processes at the Basin Scale. *Sustainability* **2020**, *12*, 4928.
https://doi.org/10.3390/su12124928

**AMA Style**

Chen W, Thomas K.
Revised SEDD (RSEDD) Model for Sediment Delivery Processes at the Basin Scale. *Sustainability*. 2020; 12(12):4928.
https://doi.org/10.3390/su12124928

**Chicago/Turabian Style**

Chen, Walter, and Kent Thomas.
2020. "Revised SEDD (RSEDD) Model for Sediment Delivery Processes at the Basin Scale" *Sustainability* 12, no. 12: 4928.
https://doi.org/10.3390/su12124928