The method of fuzzy mathematics makes up for the defect that the "comprehensive index method" ignores the fuzziness of the boundary of water quality classification. Because the groundwater environment system has the following characteristics.
1) There is a complex but unclear correlation between pollutants in the water environment system. Water pollution is the result of the interaction of various pollution factors, and it is a continuous, gradual and complex process, with fuzzy boundaries and fuzzy evaluation objectively.
2) When determining the water quality grading standard according to the use of water and environmental indicators, if the characteristics and uses are expressed by a single value of each factor, there are great human factors in the selection of standards, and the standards formulated by people for water quality are also objectively vague.
3) After various single and comprehensive operations, the water quality conclusion is given. Because the water quality is a continuously changing event, the conclusion given is also ambiguous (Sun Youping, 1988).
According to the above characteristics, in order to truly describe this process, in view of its fuzziness, fuzzy mathematics theory is used to deal with it, and the evaluation of groundwater quality will give more objective results.
(A) fuzzy comprehensive evaluation method
The so-called fuzzy evaluation is a method of overall evaluation of everything through fuzzy transformation according to the given evaluation criteria and measured values.
The problem of fuzzy comprehensive evaluation is actually a problem of fuzzy transformation, and its principle can be expressed by model (4-27):
B=A R (4-27)
Where: a is the factor weight.
In order to highlight the main factors of groundwater quality, all kinds of samples and factors are given weights according to their different situations in fuzzy classification standards, and the weight A is obtained. A is a row matrix of order 1×m, which is formed by processing the weight of each evaluation factor and is called input.
Weight a is related to the method and purpose of evaluation. The weight should be based on the contribution of each evaluation factor to groundwater quality. If a variety of factors affect the quality of groundwater, it should be able to reflect the synergy and antagonism between the factors. In the actual evaluation, it is difficult to give a reasonable weight because the mechanism of chemical components migration and transformation in groundwater system media is not easy to understand. Generally, the environmental quality sub-index method is used to calculate the weight a.
In order to carry out fuzzy transformation, Wi should meet the normalization requirements:
Study on theory and method of regional groundwater function sustainability evaluation
Normalization, the calculation formula is
Study on theory and method of regional groundwater function sustainability evaluation
Where: Wi is the weight of the normalized I factor.
Thus, a (1×m) weight matrix is formed: A=(W 1, W2, …, Wm).
R is a fuzzy converter, which consists of several single factor evaluation line vectors. It represents the fuzzy transformation relationship between the checked elements and the highest evaluation level, and its fuzzy relationship matrix is
Study on theory and method of regional groundwater function sustainability evaluation
B is the result of comprehensive evaluation, which is called output.
B is the required evaluation result, which is a fuzzy subset of the evaluation set and expressed in the form of 1×n line vector.
B=(μ(x 1)，μ(x2)，…，μ(xi))
Each element in the above formula is the membership degree of each factor to the evaluation grade.
μ n is calculated by simplified semi-trapezoid method, and the conversion formula is shown in Table 4-4 (Han et al., 2000).
Table 4-4 Membership μ n Calculation Formula List
sequential
To sum up, the known input and output of fuzzy converter is fuzzy comprehensive evaluation (Fu et al., 1987).
Comprehensive evaluation, that is, the synthetic operation of two fuzzy matrices A and R, adopts the (∧, ∨) type comprehensive evaluation calculation method, which is similar to ordinary matrix multiplication, except that "×" is changed to "∧" and "+"is changed to "∧". The results of compound operation show the membership degree of water samples relative to the comprehensive evaluation of various quality categories.
According to the comprehensive membership degree after evaluation, the membership degrees of all levels are compared. Among them, the grade with the largest degree of membership is the classification grade of water samples.
If Bi = Max {B 1, B2, ..., BN}, the water quality of the sample is judged as Grade I. ..
The water quality of multiple samples should be sorted from excellent to poor; For the same level of water quality, compare the membership degree of the superior level of the adjacent level of each sample, and the largest one will be ranked first; The water quality grade is different, and the back row is worse.
By applying the fuzzy comprehensive evaluation method to groundwater quality evaluation, we can get an objective comprehensive evaluation conclusion and the order of the influence degree of each component.
The limitation of fuzzy comprehensive evaluation method;
1) b = a r is obtained from "∨" and "∧". Overemphasizing the function of extreme value will inevitably lose some information provided by data and make the judgment result appear "rough". For example, when the evaluation function presents b 1=b2=…=bm, it will be given.
2) Because of the emphasis on "take the minimum and take the maximum", if every component in A is less than every quantity in R, then all the quantities in the synthetic result R will be screened out, which makes the single factor discrimination invalid, and thus the phenomenon of taking weight as the evaluation function appears.
The above situation will affect the accuracy of the assessment. In order to get a better evaluation result, according to the actual situation, "∨" and "∧" can be replaced by other operators for evaluation. Table 4-5 lists several common operator forms (Fu et al., 1987).
Table 4-5 Other Common Operator Forms
Note: A and B represent μa(x) and μ b (x) respectively; A b stands for ordinary real number multiplication; ⊕ means bounded sum operation.
If an operator is uncertain, multiple operators can be used to judge separately at the same time, and finally compare the evaluation results to determine an objective and better conclusion.
(2) Similar priority ratio method
Similarity priority ratio method is a calculation method in fuzzy mathematics, which establishes a fuzzy similarity relation according to some factors in the set composed of selected objects, and then determines the advantages and disadvantages of elements by the fuzzy matrix representing this fuzzy relation. In this way, the elements in the set can be sorted according to their advantages and disadvantages.
The fuzzy similarity matrix is constructed on the basis of Hamming distance ratio, and the similarity ranking between each partition and environmental target value is calculated by using the concept of λ cut matrix.
1. Hamming distance
dki=xk-xi (4-29)
dkj=xk-xj (4-30)
Where: xk is the standard value (environmental target value) of a certain water quality; Xi and xj are the measured averages of the two areas being compared.
2. Fuzzy similarity priority ratio
Study on theory and method of regional groundwater function sustainability evaluation
rji= 1-rij (4-32)
If rij is between (0.5, 1), it means that xi takes precedence over XJ; If rij is between (0,0.5), xj takes precedence over xi.
There are three ideal situations: if rij= 1, it means that xi obviously takes precedence over XJ; If rij=0, xj is obviously superior to Xi; If rij=0.5, the priority ratio cannot be determined, and the two options are equivalent.
3. Fuzzy similarity priority ratio matrix
Study on theory and method of regional groundwater function sustainability evaluation
4. Similarity
According to the actual situation, select a series of λ values from large to small between [0, 1] (λ is the boundary of the program to evaluate the similarity between samples and standard values), make a similarity matrix Rλ, find out the similarity between each factor and the target, and sort the elements in the order of finding λ cut matrix.
Study on theory and method of regional groundwater function sustainability evaluation
5. Comprehensive sorting
Comprehensive sorting, that is, summing the serial numbers of various elements, the smaller the serial number and the better, and vice versa.
The similar priority ratio method is used to rank groundwater quality, and the effect is good. However, it is complicated to establish fuzzy similarity relation matrix and find λ cut matrix. In order to avoid a large number of calculations, it is suggested to apply this method in the case of few samples (Hu et al., 1996).
(3) fuzzy distance sorting method
Fuzzy distance ranking method is a tedious work to establish fuzzy similarity relation matrix and find λ section matrix on the basis of similarity priority ratio method. By changing the order of the samples to be sorted, the properties of fuzzy priority matrix determined by fuzzy distance are analyzed, and a simplified method for sorting fuzzy priority relations is given.
Brief introduction of fuzzy distance sorting method;
Let us know what the given standard sample is.
B=(b 1，b2，…，bi) (4-35)
Where: I = 1, 2, …, m.
Suppose the sample sequence to be sequenced is A':
A′ 1，A′2，…，A′I，…，A′n(4-36)
Where: a ′ i = (ai1,ai2, …, aiji), 1≤i≤n, 1≤j≤m, that is, each sample consists of m indicators.
Due to the different index units of samples, the same index may be very different. In order to give full play to the role of sample indicators in comprehensive evaluation, firstly, the indicators of each sample in sample sequence A' are standardized. Then calculate the fuzzy distance between the sequence sample to be processed and the standard sample, using the following formula:
Study on theory and method of regional groundwater function sustainability evaluation
Where: d(Ai, b) is the fuzzy distance between sample Ai and standard sample b; Xik is the k-th index of the sample Ai to be sorted; Dk is the weight of the kth index of the sample, and; P is a selected constant, and when p= 1, Equation (4-37) is a weighted Hamming distance; When p=2, it is a weighted Euclidean distance.
Use formula (4-35) to calculate the fuzzy distance sequence d' between the sample a' to be tested and the standard sample b:
d(A 1，B)，d(A2，B)，…，d(Ai，B)，…，d(An，B) (4-38)
Arrange the sequence (4-36) from small to large, and get a new fuzzy distance sequence d:
d 1，d2，…，di，…，dn (4-39)
Where: d 1 < D2, ... < di...< dn.
Rearrange the sample sequence formula (4-36) according to the sequence corresponding to the samples in the sequence formula (4-39) to obtain a new sample sequence a:
A 1，A2，…，Ai，…，An (4-40)
If di=dj exists in the sequence formula (4-39), the sequence formula (4-40) is written as follows:
A 1，…，A2，…，Ai，Aj，…，An
Fuzzy distances are arranged from small to large, and the order in which the samples to be sorted appear in the distance series is the expected arrangement result. For the ranking of groundwater quality, the fuzzy distance from small to large represents the groundwater quality from good to bad.
The fuzzy distance ranking method is simpler than the similar priority ratio method, with less calculation and easy evaluation, especially when there are many samples, which shows the effectiveness of this method. However, there are still many problems to be further studied and discussed in this method, such as the selection of fuzzy distance formula and the determination of sample index weight. Because the fuzzy distance ranking method is adopted, the ranking result is related to the fuzzy distance. Therefore, the appropriate calculation formula should be selected according to the actual problems, and the advantages and disadvantages should be ranked according to the hydrogeological conditions and monitoring data in the study area. The selection of sample index weight is based on the contribution of each index to groundwater quality, and it is necessary to have a clear understanding of the influence of each factor in order to grasp the weight value.