# Robustness Spatiotemporal Clustering and Trend Detection of Rainfall Erosivity Density in Greece

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}∙year

^{−1}) that is associated with sheet and rill erosion, $R$ is the rainfall erosivity factor (MJ∙mm∙ha

^{−1}∙h

^{−1}∙year

^{−1}), $K$ is the soil erodibility factor (t∙h∙MJ

^{−1}∙mm

^{−1}), $L$ and $S$ are the slope length and steepness factors, C is a cover-management factor and P is a supporting practices factor. The second revised version of USLE, RUSLE2, introduced erosivity density ($ED$), as a measure of rainfall erosivity per unit rainfall. $ED$ was proposed in order to develop $R$ values for the USA on a monthly basis and a given location, due to the fact that $ED$ requires shorter record lengths, as 10 years lead to acceptable results; also because it allows more missing data than $R$ and is independent of the elevation.

## 2. Materials and Methods

#### 2.1. Data Acquisition and Processing

#### 2.2. Comparative Assessment of the Impact of Missing Data

#### 2.3. Temporal Trend Detection

#### 2.4. Clustering Analysis

## 3. Results and Discussion

#### 3.1. Annual and Monthly Erosivity Density Calculations

#### 3.2. Monte Carlo Procedure Results

#### 3.3. Erosivity Density Temporal Trends

#### 3.4. Erosivity Density Spatio-Temporal Clustering

## 4. Conclusions

- Incomplete pluviograph data can be used to compute $ED$ and achieve acceptable accuracy on the estimation of $R$.
- Stationarity of $ED$ was found for the majority of the selected stations in Greece.
- Three clusters of stations define areas in Greece with different temporal patterns of $ED$.
- Only the stations that are located in the rainy part of western Greece have $ED$ values that follow the seasonal cycle of precipitation that is common for the country.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Station locations in Greece used in the analysis obtained from the Greek National Bank of Hydrological and Meteorological Information.

**Figure 3.**In these figures, the different subsets of stations used in the analysis are presented with red color: (

**a**) Stations used for the temporal trend analysis; (

**b**) Stations used for the clustering analysis.

**Figure 4.**The grey lines represent the average monthly ED of all the stations used in clustering analysis. Some of these lines have no values for some months due to missing values. The red line symbolizes the mean annual ED values coming from all the stations, after smoothing by means of Local Polynomial Regression Fitting [59], as a non-parametric method to present smooth curves between the plotted variables.

**Figure 5.**Results from the Monte Carlo procedure that was employed for the assessment of missing values ratio to the computation of annual $R$ and $ED$ values. Grey bands symbolize the interquartile range (i.e., the 25th and 75th percentiles of the errors) per variable. Lower values of MAPE mean smaller error.

**Figure 6.**The above plot presents the frequency among all 30 indices used for the determination of the optimal number of clusters.

**Figure 8.**Monthly $ED$ distribution values from the emerged clusters. Black dots symbolize the monthly values per station. The blue lines are the LOESS lines that are fitted to the data per cluster. The grey areas represent the standard error of LOESS.

**Table 1.**Average statistical properties of annual and monthly ED values for the stations used in clustering analysis. SD is an abbreviation for standard deviation and CV for coefficient of variation (the ratio of the standard deviation to the mean).

ED (MJ/ha/h) | Min | Mean | Median | Max | SD | Skew | Kurtosis | CV |
---|---|---|---|---|---|---|---|---|

January | 0.36 | 1.10 | 1.08 | 2.23 | 0.43 | 0.38 | −0.58 | 0.39 |

February | 0.52 | 1.13 | 1.07 | 2.40 | 0.41 | 0.77 | 0.10 | 0.36 |

March | 0.52 | 1.10 | 1.05 | 2.37 | 0.36 | 1.06 | 1.47 | 0.32 |

April | 0.45 | 1.07 | 1.03 | 2.10 | 0.32 | 0.80 | 0.50 | 0.30 |

May | 0.37 | 1.39 | 1.30 | 2.64 | 0.44 | 0.53 | −0.13 | 0.32 |

June | 0.78 | 1.76 | 1.57 | 3.81 | 0.68 | 0.93 | 0.31 | 0.38 |

July | 1.08 | 2.19 | 1.89 | 5.45 | 0.99 | 1.30 | 1.23 | 0.45 |

August | 0.64 | 1.92 | 1.84 | 5.99 | 0.87 | 1.80 | 5.35 | 0.45 |

September | 0.84 | 1.75 | 1.57 | 3.48 | 0.67 | 0.82 | −0.25 | 0.38 |

October | 0.61 | 1.78 | 1.66 | 3.54 | 0.67 | 0.90 | 0.18 | 0.38 |

November | 0.58 | 1.68 | 1.56 | 3.74 | 0.65 | 0.53 | −0.21 | 0.39 |

December | 0.50 | 1.40 | 1.38 | 3.36 | 0.56 | 0.62 | 0.45 | 0.40 |

Annual | 1.28 | 2.89 | 2.75 | 5.51 | 1.13 | 0.60 | 0.14 | 0.39 |

**Table 2.**Location and analysis results for the stations with a common time length during 1965–1996. ID is an abbreviation for the station ID as reported in the Greek National Bank of Hydrological and Meteorological Information, WD for the Greek Water Divisions, Lon for longitude, Lat for latitude, El for elevation, MCV for mean coverage per station, p

_{adj}is the adjusted p-value from the test using the Benjamini & Hochberg method. When a star is marked, it indicates the test results in which the null hypothesis is rejected for a significance level α = 5%.

ID | Name | WD | Lon (°) | Lat (°) | El (m) | MCV (%) | Tau | p_{adj} | |
---|---|---|---|---|---|---|---|---|---|

1 | 200003 | GRABIA | GR07 | 22.43 | 38.67 | 381 | 73.4 | 0.12 | 0.612 |

2 | 200011 | LIDORIKI | GR04 | 22.20 | 38.53 | 548 | 69.2 | −0.09 | 0.612 |

3 | 200015 | PYRA | GR04 | 22.27 | 38.74 | 1137 | 74.8 | −0.11 | 0.612 |

4 | 200018 | AG. TRIADA | GR07 | 22.92 | 38.35 | 400 | 65.4 | 0.31 | 0.081 |

5 | 200021 | DISTOMO | GR07 | 22.67 | 38.43 | 458 | 60.3 | −0.02 | 0.919 |

6 | 200024 | LEIBADIA | GR07 | 22.87 | 38.44 | 176 | 56 | −0.27 | 0.132 |

7 | 200059 | BASILIKO | GR05 | 20.59 | 40.01 | 747 | 75.8 | −0.11 | 0.612 |

8 | 200092 | ELASSONA | GR08 | 22.19 | 39.89 | 276 | 71.7 | 0.02 | 0.919 |

9 | 200135 | KALYBIA | GR02 | 22.30 | 37.92 | 822 | 65.3 | 0.29 | 0.123 |

10 | 200142 | NEMEA | GR02 | 22.66 | 37.83 | 306 | 63.8 | −0.26 | 0.132 |

11 | 200144 | SPATHOBOUNI | GR02 | 22.80 | 37.85 | 150 | 48.1 | −0.08 | 0.612 |

12 | 200181 | LESINIO | GR04 | 21.19 | 38.42 | 2 | 59.9 | 0.45 | 0.055 |

13 | 200190 | POROS REG. | GR04 | 21.75 | 38.51 | 182 | 67.8 | −0.11 | 0.612 |

14 | 200243 | NEOCHORIO | GR03 | 22.48 | 37.67 | 704 | 63.2 | 0.14 | 0.595 |

15 | 200291 | A. ARCHANES | GR13 | 25.16 | 35.24 | 392 | 51.6 | 0.09 | 0.612 |

16 | 200309 | DRAMA | GR11 | 24.15 | 41.14 | 100 | 69.6 | 0.10 | 0.612 |

17 | 200311 | PARANESTE | GR12 | 24.50 | 41.27 | 122 | 66.1 | −0.46 | 0.005 * |

18 | 200346 | KATERINE | GR09 | 22.51 | 40.28 | 30 | 64.2 | −0.15 | 0.595 |

Method | KL [60] | CH [61] | Hartigan [62] | CCC [63] | Scott [64] | Marriot [65] | TrCovW [28] | TraceW [28] | Friedman [66] |
---|---|---|---|---|---|---|---|---|---|

NOC | 3 | 2 | 3 | 2 | 3 | 3 | 3 | 3 | 3 |

Value | 2.27 | 39.70 | 11.13 | 12.61 | 109.02 | 1.40E+12 | 568.30 | 27.72 | 26.67 |

Method | Cindex [67] | DB [68] | Silhouette [30] | Duda [69] | PseudoT2 [69] | Beale [70] | Ratkowsky [71] | Ball [72] | PtBiserial [73] |

NOC | 6 | 3 | 3 | 3 | 3 | 7 | 2 | 3 | 3 |

Value | 0.26 | 1.02 | 0.39 | 0.82 | 14.45 | 0.54 | 0.39 | 57.07 | 0.75 |

Method | Frey [74] | McClain [75] | Gamma [76] | Gplus [73] | Tau [73] | Dunn [77] | Hubert [78] | SDindex [79] | Dindex [80] |

NOC | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 3 | 3 |

Value | NA | 0.30 | 0.89 | 49.04 | 787.63 | 0.30 | Graphical | 1.97 | Graphical |

Method | Rubin [66] | Gap [31] | SDbw [81] | ||||||

NOC | 3 | 2 | 8 | ||||||

Value | −1.06 | −0.36 | 0.34 |

**Table 4.**Statistical properties of the mean monthly ED values of the clusters (centers of the clusters). SD is an abbreviation for standard deviation and CV for coefficient of variation.

ED (MJ/ha/h) | Min | Mean | Median | Max | SD | Skew | Kurtosis | CV |
---|---|---|---|---|---|---|---|---|

Cluster 1 | 0.97 | 1.34 | 1.35 | 1.89 | 0.31 | 0.18 | −1.44 | 0.23 |

Cluster 2 | 1.52 | 2.06 | 1.86 | 3.09 | 0.55 | 0.67 | −1.21 | 0.27 |

Cluster 3 | 1.00 | 2.09 | 2.00 | 4.01 | 0.89 | 0.79 | −0.48 | 0.43 |

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**MDPI and ACS Style**

Vantas, K.; Sidiropoulos, E.; Loukas, A.
Robustness Spatiotemporal Clustering and Trend Detection of Rainfall Erosivity Density in Greece. *Water* **2019**, *11*, 1050.
https://doi.org/10.3390/w11051050

**AMA Style**

Vantas K, Sidiropoulos E, Loukas A.
Robustness Spatiotemporal Clustering and Trend Detection of Rainfall Erosivity Density in Greece. *Water*. 2019; 11(5):1050.
https://doi.org/10.3390/w11051050

**Chicago/Turabian Style**

Vantas, Konstantinos, Epaminondas Sidiropoulos, and Athanasios Loukas.
2019. "Robustness Spatiotemporal Clustering and Trend Detection of Rainfall Erosivity Density in Greece" *Water* 11, no. 5: 1050.
https://doi.org/10.3390/w11051050