# “One Out–All Out” Principle in the Water Framework Directive 2000—A New Approach with Fuzzy Method on an Example of Greek Lakes

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## Abstract

**:**

^{+}. The use of fuzzy methods that express the functional relationships between variables in ecology and management has been gaining more ground recently. Here is attempted the inclusion of a fuzzy regression among the frequently monitored BQE (phytoplankton) and the outcome of OOAO application in six Greek lakes. The latter was determined by the comparison of four BQE indices in order to assess the extent to which BQEs might underpin the optimal/actual qualitative classification of a waterbody. This approach encompasses the uncertainty and the possibility to broaden the acceptable final EQR based on the character and status of each lake. We concluded that the fuzzy OOAO is an approach that seems to allow a better understanding of the WFD implementation and case-specific evaluation, including the uncertainty in classification as an asset. Moreover, it offers a deeper understanding through self-learning processes based on the existing datasets.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Biological Assessment Methods

- (a)
- OOAO: the lowest EQR of BQEs was attributed for the whole waterbody;
- (b)
- average: the arithmetic average of the EQRs for all BQEs was calculated and rounded to the nearest class; and
- (c)
- median: the median of EQRs for all BQEs was calculated and rounded to the nearest class.

#### 2.3. The Proposed Fuzzy Regression Model

_{A}) presented below:

_{i}” (e.g., almost one). The term a

_{i}represents the central value and the term c the semi-width (Figure 2).

_{j}is the independent variable (here, the phytoplankton index), m is the number of data, and ${\tilde{Y}}_{j}$ is the fuzzy predicted value of the dependent variable (OOAO principle) considering the jth data. In other words, the fuzzy linear regression uses fuzzy numbers as coefficients instead of crisp numbers.

_{j}(Figure 3). The observed data are crisp numbers (OOAO principle), whilst the fuzzy regression provides a fuzzy estimation of the OOAO.

_{j}). However, an uncertainty arises from the constant term (−0.0457, 0.0457). In contrast, if the model concludes to:

_{j}(OOAO) must be included into the produced h-cut of the fuzzy number ${\tilde{\mathrm{Y}}}_{j}$ (blue line in Figure 4).

_{1}, a

_{0}are the centers of the coefficients that correspond to the independent variable (phytoplankton index) and the constant term correspondingly. The terms c

_{1}, c

_{0}indicate the semi-widths of the coefficients that correspond to the independent variable and the constant term correspondingly. More analytically, the term $\left({\mathrm{a}}_{1}{x}_{j}+{\mathrm{a}}_{0}\right)-(1-h)\left({c}_{1}\left|{x}_{j}\right|+{c}_{0}\right)$ expresses the left boundary of the h-cut of the fuzzy number ${\tilde{\mathrm{Y}}}_{j}$, whilst the term $\left({\mathrm{a}}_{1}{x}_{j}+{\mathrm{a}}_{0}\right)+(1-h)\left({c}_{1}\left|{x}_{j}\right|+{c}_{0}\right)$ expresses the right boundary of the h-cut of the fuzzy number ${\tilde{\mathrm{Y}}}_{j}$.

_{1}, c

_{0,}≥ 0, where j = 0, 1,…, m

#### 2.4. Tested Scenarios and Basic Interpretation

**Scenario a**: the fuzzy regression was applied between phytoplankton, as the most sensitive and more frequently monitored BQE, and the OOAO values. The values for OOAO were generated using the unique yearly EQR values of phytoplankton, but the same values were kept for the indices of macrophytes and benthic macroinvertebrates for all years within each monitoring period (2012–2015 and 2016–2019). That is, the EQR values of these indices were extended beyond their actual monitored year (Table 2). According to the WFD, the monitoring frequency of macrophytes and benthic macroinvertebrates for operational monitoring is once every three years.

**Scenario b**: the fuzzy regression took place between phytoplankton yearly EQR values and the OOAO values generated using the yearly EQR values of phytoplankton and only the values of the indices of macrophytes and benthic invertebrates corresponding to the year they were actually monitored (Table 2).

_{i}) and semi-width (c

_{i}), then it is obvious that the other quality indices affect the OOAO. On the other hand, if the coefficients of the phytoplankton index have a high value, which is large central value and significant semi-width, then the phytoplankton index significantly affects the OOAO. To work properly, there has to be originally a wide range of EQR values from all indices included in the regression. When a lake has quality values only within an EQR class range, it “cripples” the lake, not allowing it to go beyond these values. When both EQR values in both axes are identical, the Phytoplankton index “commands” the OOAO, and in that case, the observation value is placed closest to the central fuzzy value. The boundaries designate the maximum acceptable value for the OOAO principle acknowledging every value in the dataset. That means that the final acceptable assessment value can be within the range set by the central value plus or minus the semi-width (ci, the second coefficient). This coefficient is the uncertainty created by the existing dataset at hand each time. When a boundary value falls over an existing observed value, this means that another BQE designated the range of acceptable values. It can be easily understood that the higher the x (phytoplankton) coefficient value, the more important is phytoplankton for the OOAO principle. Similarly, the larger the second coefficient is, the wider the acceptable ranges. It should also be noted that for every new added value in the dataset (i.e., values for 2020, 2021, etc.), the regression can produce a slightly differentiated equation, meaning new semi-widths and thus new acceptable EQR ranges.

## 3. Results

#### 3.1. BQEs Comparison and Status Classification Approaches

#### 3.2. Implementation of the Proposed Methodology

#### 3.2.1. Fuzzy Regression Scenario A

#### 3.2.2. Fuzzy Regression Scenario B

## 4. Discussion

^{+}. Moreover, Greece and especially its lake basins host mainly agroecosystems, where diffuse pollution cannot easily be tackled, and green infrastructure measures need more time to pay off in quality status terms [8,57]. This issue is aggravated under the Mediterranean climate prism and some specific hydromorphological characteristics (i.e., shore alterations, high water retention time). Moreover, a six-year period for RBMP cycles renewal can be considered as adequate for reporting progress in ecological point of view, but such a period can hinder trends and the effectiveness of policy measures [56] or include the possible benefits from action plans in a management point of view. The next step that goes beyond the OOAO principle, and is nowadays under consultation [56], is the use of new quality indicators which are focused on implementation shifts (i.e., pressure reduction, PoMs application) [56]. Novel reporting protocols can perhaps improve the digitalization and administrative simplification but add extra burden on MSs to follow the changes. It is unclear, though, if this change can positively impact the OOAO principle in any means.

## 5. Conclusions

- The inclusion of a fuzzy regression among the frequently monitored BQE (phytoplankton) and the outcome of OOAO (determined by the comparison of four BQE indices) application in lakes encompasses the uncertainty and the possibility to broaden the acceptable final EQR based on the character and status of each lake;
- The fuzzy OOAO is an approach that seems to allow a better understanding of the WFD implementation and case-specific evaluation, including the uncertainty in classification as an asset;
- It offers a deeper understanding through self-learning processes based on the existing datasets;
- As for the progress reporting of individual BQEs, this requires a more complete dataset to apply a statistically solid fuzzy regression.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Index | Metrics | Lake Type GR-DNL | Lake Type GR-SNL |
---|---|---|---|

HeLPhy | Total Phytoplankton Biovolume (mm^{3} L^{−1}) | 1.29 | 0.74 |

Cyanobacteria Biovolume (mm^{3} L^{−1}) | 0.01 | 0.01 | |

modNygaard Index | 1.03 | 1.11 | |

Chlorophyll a (μg L^{−1}) | 1.56 | 3.59 | |

HeLM | TIHelm | 7.14 | 7.14 |

Cmax (m) | 12.2 | 6.1 | |

HeLLBI | ASPT | 5.47 | 5.47 |

Odonata (% AC) | 16.67 | 16.67 | |

Simpson | 0.80 | 0.80 |

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**Figure 5.**Graphical representation of the fuzzy regression for the OOAO (“One Out–All Out”) principle (dimensionless) with respect to phytoplankton index (dimensionless) in case of Scenario A for three deep natural lakes: (

**a**) Vegoritis, (

**b**) Volvi, and (

**c**) Yliki.

**Figure 6.**Graphical representation of the fuzzy regression for the OOAO (“One Out–All Out”) principle (dimensionless) with respect to phytoplankton index (dimensionless) in case of Scenario A for three shallow natural lakes: (

**a**) Doirani, (

**b**) Lysimachia, and (

**c**) Ozeros.

**Figure 7.**Graphical representation of the fuzzy regression for the OOAO (“One Out–All Out”) principle (dimensionless) with respect to phytoplankton index (dimensionless) in case of Scenario B for three deep natural lakes: (

**a**) Vegoritis, (

**b**) Volvi, and (

**c**) Yliki.

**Figure 8.**Graphical representation of the fuzzy regression for the OOAO (“One Out–All Out”) principle (dimensionless) with respect to phytoplankton index (dimensionless) in case of Scenario B for three shallow natural lakes: (

**a**) Doirani, (

**b**) Lysimachia, and (

**c**) Ozeros.

**Table 1.**Limnological characteristics of the studied lakes. OL, oligotrophic; MT, mesotrophic; ET, eutrophic.

Lake | Altitude (m.a.s.l.) | Mean Depth (m) ^{a} | Maximum Depth (m) ^{a} | Lake Area (km^{2}) ^{a} | Trophic Status |
---|---|---|---|---|---|

Doirani | 142 | 4.5 | 5.5 | 32.4 * | ET |

Lysimachia | 16 | 3.5 | 7.7 | 13.0 | ET |

Ozeros | 22 | 3.8 | 6.1 | 10.4 | ET |

Vegoritis | 510 | 26.1 | 52.4 | 47.4 | MT-ET |

Volvi | 37 | 12.5 | 27.3 | 72.9 | ET |

Yliki | 80 | 20.9 | 38.5 | 21.6 | OL-MT |

^{a}Data available from the national monitoring program implemented by the Greek Biotope-Wetland Centre (EKBY), * approximately 44% of the lake area being within the territory of Greece.

**Table 2.**Frequency of monitoring for each biological quality element. P, phytoplankton; M, macrophytes; B-L, littoral zoobenthos; B-SP, sublittoral/profundal zoobenthos.

Lake | 1st Monitoring Period | 2nd Monitoring Period | ||||||
---|---|---|---|---|---|---|---|---|

2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |

Doirani | P | P, M | P, B-SP | P | P, M | P | P | |

Lysimachia | P | P | P, M, B-SP | P, B-L | P | P, M | P, B-L | P |

Ozeros | P | P | P, M, B-SP | P, B-L | P | P, B-L | P | P |

Vegoritis | P | P, M | P, B-SP | P | P, M | P | P, B-L | P |

Volvi | P | P, M | P, B-SP | P | P, M | P, B-L | P | |

Yliki | P | P | P, M, B-SP | P, B-L | P | P | P | P |

**Table 3.**Ecological quality of six Greek lakes, based on multi-metric indices for different biological quality elements. The ecological status assessed by the “One Out–All Out” (OOAO), median, and average approaches. Quality color scale and Ecological Quality Ratio values are presented. P, phytoplankton [29]; M, macrophytes [30]; B-SP, sublittoral/profundal zoobenthos [32]; B-L, littoral zoobenthos [33].

Lake Type | Lake | Monitoring Period | P | M | B-SP | B-L | OOAO | Median | Average |
---|---|---|---|---|---|---|---|---|---|

Deep | Vegoritis | 1st (2012–2015) | 0.66 | 0.75 | 0.54 | 0.54 | 0.66 | 0.65 | |

2nd (2016–2019) | 0.64 | 0.62 | 0.69 | 0.62 | 0.64 | 0.65 | |||

Volvi | 1st (2012–2015) | 0.45 | 0.70 | 0.41 | 0.41 | 0.45 | 0.52 | ||

2nd (2016–2019) | 0.46 | 0.71 | 0.44 | 0.44 | 0.46 | 0.53 | |||

Yliki | 1st (2012–2015) | 0.77 | 0.69 | 0.34 | 0.48 | 0.34 | 0.59 | 0.57 | |

2nd (2016–2019) | 0.75 | 0.75 | 0.75 | 0.75 | |||||

Shallow | Doirani | 1st (2012–2015) | 0.56 | 0.77 | 0.69 | 0.56 | 0.69 | 0.68 | |

2nd (2016–2019) | 0.57 | 0.77 | 0.57 | 0.67 | 0.67 | ||||

Lysimachia | 1st (2012–2015) | 0.59 | 0.59 | 0.39 | 0.51 | 0.39 | 0.53 | 0.51 | |

2nd (2016–2019) | 0.53 | 0.42 | 0.63 | 0.42 | 0.53 | 0.52 | |||

Ozeros | 1st (2012–2015) | 0.71 | 0.45 | 0.53 | 0.49 | 0.45 | 0.51 | 0.55 | |

2nd (2016–2019) | 0.72 | 0.62 | 0.52 | 0.52 | 0.62 | 0.62 |

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**MDPI and ACS Style**

Latinopoulos, D.; Spiliotis, M.; Ntislidou, C.; Kagalou, I.; Bobori, D.; Tsiaoussi, V.; Lazaridou, M. “One Out–All Out” Principle in the Water Framework Directive 2000—A New Approach with Fuzzy Method on an Example of Greek Lakes. *Water* **2021**, *13*, 1776.
https://doi.org/10.3390/w13131776

**AMA Style**

Latinopoulos D, Spiliotis M, Ntislidou C, Kagalou I, Bobori D, Tsiaoussi V, Lazaridou M. “One Out–All Out” Principle in the Water Framework Directive 2000—A New Approach with Fuzzy Method on an Example of Greek Lakes. *Water*. 2021; 13(13):1776.
https://doi.org/10.3390/w13131776

**Chicago/Turabian Style**

Latinopoulos, Dionissis, Mike Spiliotis, Chrysoula Ntislidou, Ifigenia Kagalou, Dimitra Bobori, Vasiliki Tsiaoussi, and Maria Lazaridou. 2021. "“One Out–All Out” Principle in the Water Framework Directive 2000—A New Approach with Fuzzy Method on an Example of Greek Lakes" *Water* 13, no. 13: 1776.
https://doi.org/10.3390/w13131776