Evolution of the water quality of the Castanhão Reservoir
The temporal evolution of the trophic state of the Castanhão reservoir from April 2014 to December 2021 is shown in Fig. 2. This diagnosis of eutrophication of the water body was determined by the index of Toledo Júnior et al. (1983).
The results indicated the condition of eutrophication of the reservoir most of the time, with few dates presenting the condition of non-eutrophication. This scenario is mainly a reflection of the climatic conditions of the period, as the reservoir is located in a region that experienced a prolonged drought during the study period (Pontes Filho et al., 2020). Wiegand et al. (2021) observed that the drought was responsible for raising the TSI of reservoirs in NEB during this period. This was mainly due to the gradual reduction in the volume stored, as this increases the internal concentration of pollutants (Rocha Júnior et al., 2018), and the occurrence of heavy rains after dry periods (Santos et al., 2014), which mobilize the nutrients stored in the basin during the absence of precipitation, increasing the phosphorus supply in the reservoirs (Lisboa et al., 2020). These impacts are consistent with the results of Raulino et al. (2021), who also pointed to the decline in the volume of the Castanhão reservoir as the main mechanism for the deterioration of its water quality in the period studied.
In general, this trend of degradation of water quality due to climatic conditions is observed in the tropical semiarid reservoirs (Chaves et al., 2013; Lima et al., 2015; Raulino et al., 2021, 2022; Lima Neto et al., 2022; Rocha & Lima Neto, 2022).
Membership function
Figure 3 shows the representation of the trophic classification system of Toledo Júnior (1990) using triangular and trapezoidal fuzzy numbers. It is possible to observe perfect compatibility between the triangular and trapezoidal membership functions and the key points defined from the trophic classification system used, confirming the choice of the respective types of membership functions and the suitability of the selection criterion.
Feng & Ling (2014) suggest the conditions of the study area as a criterion for formulating the membership functions. However, this is an inadequate criterion, since, as the central objective of the FSE is to consider the uncertainties of the trophic classification system, it is essential to consider as a criterion information on trophic ranges (average values of ranges and transitions), according to the vast majority of studies on this subject (e.g. Lu & Lo, 2002; Taheriyoun et al., 2010; Lin & Huang, 2015; Chen et al., 2021) and the results found in this study.
Figure 3 also shows that the trophic indicators TP and Chl-a have the largest tolerance interval corresponding to the eutrophic state, whereas for the trophic indicator SD it corresponds to the mesotrophic state. This is because these trophic degrees show the greatest imprecision in the trophic ranges of the respective trophic indicators (Chang et al., 2001). Chang et al. (2001) point out that the adequate representation of these inaccuracies is essential for the fuzzy mechanism for generating the global TSI to be consistent with reality.
All trophic states are characterized by transitions that simultaneously consider the two trophic states that generate them. As a result, we have a classification mechanism that considers inputs from each trophic indicator as belonging to all trophic levels to a certain degree. In this sense, the triangular and trapezoidal membership functions were able to characterize the uncertainties of the trophic state transitions (Chang et al., 2001). This result confirms the ability of the membership function to represent uncertainties of the classification system (Lu et al., 1999).
It is important to point out that the SD indicator decreases its trophic degree from left to right, contrary to the TP and Chl-a parameters. This behavior, however, does not generate divergences in the classification mechanism, as it considers independent inputs (Taheriyoun et al., 2010).
Another important point is that the triangular function is particularly useful in situations of sparse data and that have a wide range of variation (Wang & Ran, 2019), as is the case of poor monitoring of water quality of tropical semiarid reservoirs (Wiegand et al., 2021). It should be noted that, from the results, the trapezoidal function also has this capability. Therefore, such functions are useful tools to represent the TP concentration, being fundamental for the performance of the FSE model. Thus, the membership functions demonstrated the capacity to adequately represent the uncertainties of the transitions of the trophic classification of Toledo Júnior (1990).
Judgment matrix and weighted vector
The results of the consistency analysis of the J initial judgment matrix are (Table 3): \({{\lambda }}_{max}\) = 3.0092, CI = 0.0046, RI = 0.5800 e CR = 0.79%. Based on the criteria of Chen et al. (2021) (CR < 10%,), the initial judgment matrix meets the required consistency. Therefore, there was no need to adjust the values of the J matrix.
After the normalization processes, the result of the weighted vector was reached through the AHP process:
$$\text{W}\text{ }\text{=} \left[\begin{array}{ccc}\text{0}\text{.}\text{5390}& \text{0}\text{.}\text{2972}& \text{0}\text{.}\text{1638}\end{array}\right]$$
16
Eq. (1) expresses the relative importance of the trophic indicators used in this study. It is observed that the highest weight corresponds to the TP concentration, so the level of eutrophication is determined mainly as a function of this trophic indicator (Taheriyoun et al., 2010). On the other hand, SD has the lowest relative importance for eutrophication in this study. These results are consistent with studies that used the AHP method to weigh the effects of water quality parameters on the eutrophication phenomenon (Lu & Lo, 2002; Chen et al., 2021).
The vast majority of studies on water quality in tropical semiarid region reservoirs demonstrate that the phenomenon of eutrophication is conditioned by phosphorus (e.g. Moura et al., 2020; Raulino et al., 2021; Lima Neto et al., 2022). Therefore, the relative importance of TP concentration for the phenomenon is consistent with reality. On the other hand, studies indicate that the SD can be influenced by the high turbidity resulting from mineral material in suspension and not only by the density of planktonic organisms, which would impair the trophic classification of the reservoir (Cunha et al., 2013). Thus, its lower relative importance of the SD tends to reduce the errors in the characterization of of eutrophication (Toledo Júnior, 1990).
In general, these results are consistent with other studies of this nature, which attested to the AHP method to determine the relative importance of the trophic indicators on the eutrophication (Lin & Huang, 2015; Chen et al., 2021).
Comparative analysis of reservoir trophic state modeling
Table 5 shows the comparison of the trophic states generated by the Toledo index and by the FSE. The objective of this comparison was to validate the FSE for tropical semiarid reservoirs. Although differences are observed between the TSI and the FSE (highlighted in red), in general, the results are compatible.
The two models indicated different classifications in eight dates of the historical series without relevant impacts. Two of these dates showed negligible divergences regarding decision-making because they indicated no eutrophication (≤ mesotrophic). On the other hand, although the two models converged in terms of indicating eutrophication, the FSE indicated eutrophication to a greater degree than the TSI on the other six of these. Furthermore, the FSE was the only one that pointed out the hypereutrophic level.
The more pessimistic results of the FSE model are justified by the high TP concentrations, which have greater weight in determining the trophic state of the water body in the structure of the developed model. This result is consistent with studies carried out in the tropical semiarid reservoirs, which point to phosphorus as the causative agent of the eutrophication (e.g. Moura et al., 2020; Wiegand et al., 2021). Taheriyoun et al. (2010) point out that one of the most important characteristics of the FSE method is precisely the consideration of different weights for the trophic indicators that make up the fuzzy relation matrix. The conventional TSI model has equal weights for its trophic indicators, not being able to properly weight the relative influence of each water quality parameter on the final trophic degree (Chen et al., 2021). As a consequence, the effects of Chl-a concentration and SD are amplified in the model by Toledo Júnior et al. (1983). These results are in agreement with the study by Lu et al. (1999). These authors also point out that the unlimited amount of influence that an individual factor can exert on the index and the failure to weight each factor, which can create inconsistent results, are disadvantages for traditional TSI models.
In May 2017 and in the last three dates in Table 5, important divergences are observed in the trophic state, with the Toledo index indicating eutrophication of the water body and the FSE pointing no eutrophication. It should be noted that other studies also pointed out divergences between the FSE and the traditional TSI model regarding the possibility of eutrophication (e.g. Lu et al., 1999; Wang & Ran, 2019).
The four dates present, respectively, the matrices of results [0.10 0.41 0.49 0], [0 0.38 0.62 0], [0 0.38 0.62 0] e [0 0.31 0.69 0]. Although divergences are observed between the trophic states indicated by the two models, the levels of membership of the eutrophic state in the three dates above 0.30 indicate that the FSE model is also able to capture the possibility of eutrophication of the reservoir with a relatively high degree of certainty. These results and similar observations were pointed out in studies by Wang & Ran (2019). In this sense, FSE model offers broader results and compensates for the deficiency inherent in traditional (determistic) TSI models.
Table 5 - Comparison of the trophic state generated by the Toledo index and by the FSE.
Date
|
TP (µg.L-1)
|
Chl-a (µg.L-1)
|
SD (m)
|
TSI
|
FSE
|
12/04/2014
|
48.00
|
4.49
|
3.80
|
Oligotrophic
|
Mesotrophic
|
03/04/2015
|
64.00
|
13.63
|
1.60
|
Mesotrophic
|
Mesotrophic
|
05/21/2015
|
87.00
|
2.84
|
3.00
|
Mesotrophic
|
Oligotrophic
|
08/11/2015
|
74.00
|
18.88
|
1.70
|
Mesotrophic
|
Mesotrophic
|
01/21/2016
|
90.00
|
19.70
|
1.70
|
Eutrophic
|
Eutrophic
|
07/27/2016
|
136.00
|
52.33
|
1.20
|
Eutrophic
|
Eutrophic
|
10/19/2016
|
102.00
|
34.38
|
1.30
|
Eutrophic
|
Eutrophic
|
01/26/2017
|
236.00
|
22.68
|
1.20
|
Eutrophic
|
Hipereutrophic
|
04/27/2017
|
64.00
|
54.08
|
1.20
|
Eutrophic
|
Mesotrophic
|
07/25/2017
|
120.00
|
51.80
|
1.00
|
Eutrophic
|
Eutrophic
|
11/07/2017
|
103.00
|
45.00
|
0.90
|
Eutrophic
|
Eutrophic
|
01/10/2018
|
92.00
|
45.92
|
0.70
|
Eutrophic
|
Eutrophic
|
04/10/2018
|
190.00
|
56.14
|
0.80
|
Eutrophic
|
Hipereutrophic
|
07/11/2018
|
154.00
|
24.63
|
1.30
|
Eutrophic
|
Eutrophic
|
10/16/2018
|
154.00
|
7.09
|
0.60
|
Eutrophic
|
Eutrophic
|
01/08/2019
|
189.00
|
93.56
|
0.60
|
Eutrophic
|
Hipereutrophic
|
04/02/2019
|
225.00
|
78.72
|
0.50
|
Eutrophic
|
Hipereutrophic
|
07/09/2019
|
205.00
|
29.50
|
1.00
|
Eutrophic
|
Hipereutrophic
|
10/08/2019
|
83.00
|
33.26
|
0.90
|
Eutrophic
|
Eutrophic
|
05/19/2020
|
219.00
|
5.61
|
1.60
|
Eutrophic
|
Hipereutrophic
|
08/18/2020
|
103.00
|
12.78
|
2.20
|
Mesotrophic
|
Mesotrophic
|
11/19/2020
|
66.00
|
9.64
|
2.40
|
Mesotrophic
|
Mesotrophic
|
03/17/2021
|
46.00
|
23.59
|
1.20
|
Mesotrophic
|
Mesotrophic
|
06/15/2021
|
93.00
|
15.80
|
1.40
|
Eutrophic
|
Mesotrophic
|
09/14/2021
|
64.00
|
25.20
|
1.30
|
Eutrophic
|
Mesotrophic
|
12/21/2021
|
70.00
|
23.57
|
1.45
|
Eutrophic
|
Mesotrophic
|
Eutrophication Index (EI) vs Result Matrix B
Table 6 highlights the eutrophication index (EI) values and their respective trophic levels at which trophic status and EI are incoherent. It is observed, for example, that the EI of the oligotrophic state of the date 05/21/2015 is greater than the EI of the mesotrophic state of the date 12/04/2014. This problem was also observed in studies by Lu et al. (1999) and Lu & Lo (2002). This type of situation points to the failure of the eutrophication index to rank eutrophication levels and, consequently, as an inadequate tool to help manage eutrophication. Apparently, the reason for the inadequacy of the EI to quantify the degree of eutrophication is the arbitrary choice of its weights (Lu et al., 1999; Lu & Lo, 2002; Taheriyoun et al., 2010).
Table 6
EI values and their corresponding trophic degrees.
Data
|
TP
(µg.L− 1)
|
Chl-a (µg.L− 1)
|
SD
(m)
|
EI
|
FSE
|
\({\text{b}}_{\text{H}}\)
|
\({\text{b}}_{\text{E}}\)
|
\({\text{b}}_{\text{M}}\)
|
\({\text{b}}_{\text{O}}\)
|
12/04/2014
|
48.00
|
4.49
|
3.80
|
1.65
|
Mesotrophic
|
0
|
0.05
|
0.55
|
0.40
|
05/21/2015
|
87.00
|
2.84
|
3.00
|
1.82
|
Oligotrophic
|
0
|
0.28
|
0.26
|
0.46
|
08/11/2015
|
74.00
|
18.88
|
1.70
|
2.63
|
Mesotrophic
|
0
|
0.30
|
0.54
|
0.16
|
10/19/2016
|
102.00
|
34.38
|
1.30
|
2.63
|
Eutrophic
|
0
|
0.63
|
0.37
|
0
|
01/21/2016
|
90.00
|
19.70
|
1.70
|
2.90
|
Eutrophic
|
0.08
|
0.51
|
0.44
|
0.16
|
07/27/2016
|
136.00
|
52.33
|
1.20
|
3.02
|
Eutrophic
|
0.11
|
0.80
|
0.09
|
0
|
07/25/2017
|
120.00
|
51.80
|
1.00
|
2.99
|
Eutrophic
|
0.08
|
0.83
|
0.09
|
0
|
11/07/2017
|
103.00
|
45.00
|
0.90
|
2.90
|
Eutrophic
|
0.07
|
0.76
|
0.17
|
0
|
01/10/2018
|
92.00
|
45.92
|
0.70
|
2.96
|
Eutrophic
|
0.19
|
0.58
|
0.23
|
0
|
10/16/2018
|
154.00
|
7.09
|
0.60
|
3.02
|
Eutrophic
|
0.31
|
0.39
|
0.30
|
0
|
05/19/2020
|
219.00
|
5.61
|
1.60
|
2.84
|
Hipereutrophic
|
0.54
|
0
|
0.22
|
0.24
|
Table 6 demonstrates that the membership levels correct the divergences between the EI and their respective trophic level. Therefore, as an alternative to the IE, the membership levels associated with the trophic states can be used in the FSE, specifically the result matrix B. Other studies, although they did not explicitly discuss this problem, pointed to solutions in this direction (e.g. Zou et al., 2006; Lu et al., 2010; Taheriyoun et al., 2010; Wang & Ran, 2019).
Zou et al. (2006) characterized the eutrophication of reservoirs based on the Max operator, making it possible to perform a hierarchy between the different reservoirs studied based on the values of the levels of membership associated with the trophic degrees. Wang & Ran (2019) emphasized that membership levels can prevent the occurrence of inconsistent values of the degree of eutrophication.
Therefore, it is confirmed that the global classification of the trophic level must be performed with the Max operator applied to the result matrix B (Eq. 14), however the level of pertinence associated with the global trophic state must have been adopted as an additional tool instead of the EI, especially for hierarchical purposes of the degree of eutrophication of the water body or between water bodies.
Traditional Trophic State Index vs Fuzzy Synthetic Evaluation
The convergence of the results, in general, indicates that the traditional TSI model is satisfactory to characterize the eutrophication. However, it is necessary to highlight two additional advantages of the FSE as a tool for managing reservoir eutrophication.
Initially, the FSE indicates how much the condition of the water quality of the reservoir fits the concept of eutrophication in terms of a fuzzy set, and the specific membership value generated by the Max operator allows an easy quantitative hierarchy of a set of reservoirs in terms of the degree of eutrophication. A direct effect, for example, is on the allocation of resources for the management of the reservoir and priorities for the application of measures to mitigate eutrophication (Chen et al., 2021).
Second, the results of the evaluation matrix B express how much the body is contained in the different trophic states of the adopted classification system. This indication is crucial to indicate the importance of using the FSE, as it consists of quantifying the uncertainties generated by the transitions of pairs of trophic levels, which is the major inconsistency generator of traditional TSI models (Lu et al., 2010). Furthermore, reservoirs have a large degree of variability in their response to a trophic indicator, such as nutrient concentration, and this leads to uncertainties in the trophic state response (Novajan et al., 2019). Since, the membership levels in terms of the different trophic levels also face this uncertainty (Zou et al., 2006).
Therefore, the information generated by the FSE method is more reliable and adequate for the proper management of water quality in reservoirs due to the uncertainties that permeate the eutrophication classification system of water bodies.