Study on Eutrophication Status Evaluation

Abstract: Given
that results of traditional assessing methods presently used are extreme values
and of poor resolution, a set pair analysis based on triangular fuzzy number is
established and a new method of calculating weight is constructed by using the weighted
average method. The case study shows that the method presents the evaluating
grade as a confidence interval which makes it more feasible and suitable to
practical conditions.

Keywords: set pair analysis;
triangular fuzzy number; super-standard multiple weight method; confidence
interval; lake eutrophication.

1       
Introduction  

Sustainable
management of water resources requires a more detailed understanding of
spatiotemporal trends and differences in hydrological parameters; even though
large improvements in water quality have been achieved as a result of tieless
efforts to reduce eutrophication, nutrient enrichment continue to be the main
cause of water pollution for many freshwater ecosystems in the developing world
[1]. Considering that the demand for freshwater resources is expected to
increase, protecting diminishing water resources has become one of the most important
water management and pollution control issues [2].Water quality assessment is
the back-bone of lake water management and pollution control, it provides water
resource managers with much needed information to understand the state of the
water bodies in order to make sound decisions for proper management of this
vital resource.

Many
credible water quality assessment methods have been designed and used,
including; single factor evaluation method, comprehensive water quality
evaluation method, fuzzy evaluation method, grey evaluation method, artificial
neural network method, parameter method, biological evaluation method and
nutritional status evaluation method, set pair analysis method etc [3]. All of
these methods were widely used in water quality evaluations of individual water
bodies, and of watersheds. These existing methods were used to provide
information that formed an important foundation for promoting the rational
development, planned use, and protection of water resources [4].  In recent years the comprehensive evaluation
methods are extensively used in China because the methods allows a more
thorough investigation of water quality and a more continuous depiction of the
eutrophication processes, however, 
several problems are inherent in these methods. These methods include
the fuzzy comprehensive evaluation which has been proved effective in solving
problems of fuzzy boundaries and controlling the effect of monitoring errors on
assessment results [5]. However, there are still some limits when applying fuzzy
comprehensive evaluation method to water quality assessment; For example, when
the method emphasizes extreme value action, more information is lost and the
scientific character of weight value is not clear etc [6]. Set pair analysis
model has higher objectivity and reliability compared with the traditional
fuzzy comprehensive evaluation model [7]. Nevertheless; it does not deal with
the weight of the evaluation index [8]. 
Weight plays a key role in the comprehensive evaluation mathematical
model, reflecting the position and role of each index in the procedure of
comprehensive decision making, and directly influencing the result of the
comprehensive evaluation. For the same measured data, the low content and high
standard allowable concentration largely affect the pollution comparatively
[9].  It is difficult to establish a
unified evaluation model, which promotes the further study on the water quality
assessment methods [10]. 

This
paper presents a new method based on confidence interval in evaluation of lake eutrophication
that uses the resolving power of the set pair analysis model (SPA) and the
triangular fuzzy number (TFN) to deal with certain-uncertainty information in
lake assessment. It clearly defines the scientific character of weight value
using the weighted average method and furthermore gives the final evaluating
grade as a confidence interval which has a commonly held interpretation in the
scientific community and more widely has advantages of high resolution and
information utilization.

2       
Methodology

The improved set pair analysis model based on triangular fuzzy
number (ISPA (TFN)) was
established as follows

2.1  Determination
of Connection Degree Formula

Set
pair analysis is based on the three-element
connection number as a-
identity, b- discrepancy, c-contrary connection degree as shown below.

                                 (1)                                 

In its practical
application, there are some problems that it is rough only to divide
state-space of research object into three [11] which cannot describe the
problem explicitly. Therefore, the basic expression of the degree of couplet
should be extended at different levels and at the same level. A multi-element connection
number is formed; whose expression is as shown below [12]

                    
(2)

In order to set up a
five-element connection number in relation to I-V grades evaluation standard
commonly used in lake assessment the b-
discrepancy was expanded into

– partial
identity in discrepancy,

– entirety
discrepancy,

– partial
contrary in discrepancy,

 – contrary.
The partial identity in
discrepancy is divided into identity, the partial identity in discrepancy and
the entirety discrepancy according to a certain proportion. The entirety
discrepancy is divided into the partial identity in discrepancy, the entirety
discrepancy and the partial contrary in discrepancy according to a certain
proportion. The partial contrary in discrepancy is divided into the entirety
discrepancy, the partial contrary in discrepancy and contrary. Using the above
method can deal with the difficult problem of measure scale too wide [13] as follows.

                              (3)

Where:

 is the coefficient of partial identity in
discrepancy,

– is the
coefficient of entirety discrepancy,

 – is the coefficient of partial contrary in
discrepancy,

– is the
coefficient of contrary.

2.2  Determination
of coefficient of the discrepancy degree
based on triangular
fuzzy number

Adopting analysis technique determines
the coefficients of discrepancy degree i.e

 = 0.5,

= 0,

= -0.5.
Thereby, the coefficients of connection form a group of sequence: 1, 0.5, 0,
-0.5, -1 which are continuous in [-1, 1].
So the coefficients of connection have
the minimum value

the optimum value

 and the maximum
value

,The coefficient of discrepancy degree basing on the
triangular fuzzy intervals under ?-cut can be constructed as follows triangular
fuzzy number TFN

.[14]. The
triangular fuzzy numbers of the coefficient of partial dissimilarity, the
coefficient of the middle difference degree, the coefficient of partial
contrast coefficient and the coefficient of the coefficient of partial
opposition are determined [15] the TFN are constructed as follows the partial
difference coefficient with triangular fuzzy number’s à (

) = (0, 0.5, 1), Triangular fuzzy number
difference degree coefficient in  Ã (

) = (-0.5, 0, 0.5) and
Triangular fuzzy number coefficient partial contrast degree of à (

)= (-1, -0.5, 0).

If the confidence level ? is
given, we can obtain the confidence interval under the confidence level ? as
follows [16]:

à a(

) = [0.5a, -0.5a+1]

à a(

) = [0.5

 – 0.5, – (0.5

 -0.5)]                                      (4)

à a(

) = [0.5

 – 1, – 0.5a]

If
a=1,
then à a(

) =
0.5, Ã a(

) =
0, Ã a(

) = – 0.5
which is identical with the results of analysis technique

2.3  Determination of the connection degree formula based on triangular fuzzy intervals

Adopting
the triangle membership function to calculate connection components, and then
to set up a connection degree formula [17] The hierarchical connection number expression based on
the confidence interval is obtained by using the coefficients of the triangular
fuzzy numbers in the hierarchical relation as follows [16].

 

 

           (5)

 

 

 

 

Where,

is the m-th evaluation index,

;

is the k-th evaluation sample,

2, …, K;

is the k-th evaluation
sample value;

,

,

,

and

.

2.4          
Determination
of connection degree matrix

The connection degree value [

,

] is used to establish a connection degree matrix R as
follows.

                  (6)

2.5          
Determination
of every evaluation index weight.

In the weighted average method, categories of
environmental factors are assigned a weight which determines how much that environmental
factor counts towards your final grade by taking the averages of a collection
of factors, some factors are assigned a greater “weight”, or importance than
others. The weighted average method considers two parts equally, one of which
is the super-standard extent, and the other is the difference among water
quality levels. In the former part, the super-standard of every index at each
evaluating object is calculated; the larger the amount of pollution, the
greater the weight; in the latter part, differences among levels of water
quality standard are taken into consideration, the worse the level, the greater
the weight [18].

The computational equation for the
index weight is:

Where,

 represents the measured value of index i and

 is the arithmetic mean of index i in each grading representative value;

 is the typical value of index i in each grading standard. In order to
make the compositional operation, the weight of each single factor should be
normalized as follows [9]:

 

                                                             

(8)
Where,

 is the normalized weight of the evaluation
index i.         

2.6             
Determination of a
comprehensive assessment analysis

Adopting the comprehensive method of the addition weighting carry out
the system comprehensive estimation, as follows [19]:

Where,

                             (9)

The result of comprehensive estimation

is also
a confidence interval number, and

2.7             
Calculation of the
grade of lake eutrophication Status

According to the relation frame of identity,
discrepancy and contrary, the projection function of the grade

and results

of comprehensive estimation are established as follows [20]:

                             (10)

By using the projection function equation 10,
the grade of comprehensive estimation can be determined. Let the confidence
level of interval number be a and

3       
Case study

Using Qinghai lake as an example, a
case study is done to examine the feasibility and effectiveness of
the proposed method. Table 1 shows the sample
values of the evaluation index set A=

 the sample indexes
values of fifteen lakes in China and Table
2 shows the evaluation standard set B=

.

Where each evaluated index threshold value of “oligotropic”,
“oligo-meso”, “mesotrophic”, “meso-eutro”, and “eutrophic”, respectively.
Similarly, for

, the graded connection degree can be defined [10] as
follows.a??????? ] 1 ,0[Îa? ) 1(mS ? ) 2(mS ? ) 3(mS ? ) 4(mS ? ) 5(mS ????????????????x???????????m??m??????k ??k ?????? M

Table 1 Surveyed data of evaluation water quality in China lakes

Lakes

TP
(mg/m3)

COD Mn
(mg/L)

SD
(m)

TN
(mg/m3)

BIO
(104/L)

Qinghai lake

20

1.40

4.50

0.22

14.60

Chao lake

30

8.26

0.25

1.67

25.30

Waihaizhong lake

40

3.53

0.92

0.87

916.00

Waihaibei lake

56

3.37

0.87

1.08

945.00

Wuhan east lake

105

10.70

0.40

2.00

1913.70

Hangzhou west lake

130

10.30

0.35

2.76

6920.00

Tianchi lake

23

4.05

1.20

0.62

51.63

Erhai lake

34

2.11

0.30

0.49

22.36

Hulun lake

80

8.29

0.50

0.13

11.60

Fuxian lake

20

1.61

7.03

0.21

19.00

Hongze lake

100

5.50

0.30

0.46

11.50

Tai lake

20

2.83

0.50

0.90

100.00

Dianchi lake

20

10.13

0.50

0.23

189.20

West lake

33

3.70

1.60

0.50

8.01

Ping lake

177

51.00

0.41

2.86

7060.00

Table 2 Standards of the Eutrophication of
the lakes

TP
(mg/m3)

TN
(mg/m3)

CODMn
(mg/L)

BIO
(104/L)

SD
(m)

Trophic state
 

<1 <0.02 <0.09 <4 ?37.00 Oligotrophic (grade I) 4 0.06 0.36 15 12.00 Oligo-meso (grade II) 23 0.31 1.80 50 2.40 Mesotrophic (grade III) 110 1.20 7.10 100 0.55 Meso-eutro (grade IV) >660

>4.60

>27.10

>1000

>0.17

Eutrophic (grade V)

 

Table 3 Connection degree and weight vector for indexes of Qinghai Lake

Evaluation index

L

R

weight vector

-0.2961

-0.0461

0.1267

-0.2361

0.0139

0.1941

-0.0833

0.1667

0.4364

-0.1950

0.0550

0.1797

0.1455

0.5261

0.0631

 

Given that set pair is defined as a pair that consists of two
interrelated sets, putting together set A and B to compose set pair H with
respect to the problem W [21]. The
connection degree formula of the evaluation index can be obtained using
Equation (3) using confidence level ? as 0.75; we can obtain the confidence
interval of the coefficient of discrepancy degree and contrary degree based on
the triangular fuzzy intervals. Therefore, the connection degree value of the
evaluation index sample values

can be obtained using equation
(5), then using equation (6) the connection degree matrix R is constructed by

 as shown in table 3. The weight vector

of evaluation can be obtained
using weighted average equations (7 and 8 ) as shown in table 3 and using the equation (9 and
10) the grade of lake eutrophication status assessment results are obtained and
the comparison shown in table 4 below .

Table 4 Comparison of ISPA (TFN)
with ISPA method

Lakes

Lower
limit

Upper
limit

Grades      ISPA (TFN)

Grades
ISPA

Qinghai lake

2.7854

3.2911

III

III

Chao lake

3.9767

4.4141

IV

IV

Waihaizhong lake

3.8204

5.0000

IV

IV

Waihaibei lake

3.8309

5.0000

IV

IV

Wuhan east lake

4.3612

5.0000

IV

IV

Hangzhou west lake

4.7128

5.0000

V

V

Tianchi lake

3.2902

3.7254

III

III

Erhai lake

3.2126

3.8190

III

III

Hulun lake

2.0186

4.8377

III

III

Fuxian lake

2.6296

3.1623

II

II

Hongze lake

3.8488

4.2981

IV

IV

Tai lake

3.7382

4.3756

IV

IV

Dianchi lake     

3.9974

4.9133

IV

IV

West lake

3.5171

3.9438

III

III

Ping lake

4.8894

5.0000

V

V

 

An
improved SPA evaluation model of lake eutrophication considers both single and
multiple criteria. It can be seen from Table 4 that evaluation results of the
improved SPA are coherent to the ISPA of Reference [10].  Given the fact that both the difference among
water quality levels and the proportion of the pollutant over standard are
taken into consideration
shows that ISPA (TFN) is more feasible and informative,
compared to traditional methods the proposed method is easier to interpret,
understand and provides a strong scientific basis for decision making in
Sustainable management of lakes.

4       
Conclusion

(1)               
The
improved set pair analysis based on triangular fuzzy numbers enriches the
traditional set pair analysis method, which is relatively simple and accurate,
and has a wide application in the evaluation of lake eutrophication. The traditional evaluation method can
only get a rough assessment of the evaluation grade, and can’t further
distinguish between the same grades. The set pair analysis based on triangular
fuzzy number cannot only give the specific evaluation grade,
but also obtain a confidence interval. 

(2)               
 The
improved set pair analysis based on triangular fuzzy intervals the confidence interval
results are different when the confidence level a takes different values
between 0 and 1. The results are not a confidence interval but a precise value
while a=1.

(3)               
The
super-standard multiple weight method calculates the weight coefficient by the
standard value of quality level of monitor, the results can reflect the degree
of contribution to subordinate grade of the monitor data of water environment
factors accurately. It is closer to the actual result of the weight than the
traditional method. This method takes into account the portions of water
environment factors that may have uneven representation, and account for them
by making the final assessment result reflect a more balanced and equal
interpretation of the monitor data accurately.