Flash flood risk evaluation based on variable fuzzy method and fuzzy clustering analysis

Abstract

Flash flood is one of the most significant natural disasters in China, particularly in mountainous area, causing heavy economic damage and casualties of life. For numerous small hilly basins that need flash flood prevention and control with limited funds, it is necessary to give a priority order or to determine which basin needs to be harnessed firstly. Flash flood risk assessment is critical to an efficient flash flood management. Among many flash flood risk evaluation methods in literatures, variable fuzzy method (VFM) was chosen in this paper. To verify the results of VFM, fuzzy clustering analysis (FCA) is also used. First, taking Licheng county with 119 small basins in China as an example, 9 indexes were identified among index system, based on disaster-breeding environment (or underlying surface conditions) of small basin in hilly region. Risk levels are divided into three grading levels such as high, medium and low. Second, VFM was introduced, and the flash flood risk grade eigenvalue (H) of each small basin was calculated. The results show that no small basin belongs to high risk level, 14 basins belong to low risk level, and the remaining 105 small basins belong to medium risk level. Third, FCA was used to verify the result of VFM. The results of two methods show that they are nearly in consistence. This paper shows that VFM is feasible for flash flood risk evaluation. Finally, the priorities for flash flood mitigation of 119 small watersheds in Licheng county are mapped out, which will provide effective help for flood disaster mitigation of small basin.

Keywords

Variable fuzzy method fuzzy clustering analysis flash flood risk disaster-breeding environment small basin

1. Introduction

Flash flood is one of the worst natural disasters worldwide. It can cause extensive disruptions to property, infrastructure and human lives [1 –3]. At present, flash flood disaster has the characteristics of high frequency, rapid burst, and large losses. In China, there are more than 530,000 small basins in 2,058 counties where mountain flash flood disasters need to be mitigated [3]. In practice, with limited funds for flash flood mitigation, the priorities of small watersheds for flash flood prevention and control are also needed for an efficient flash flood management. Therefore, the flash flood risk evaluation of small basins is needed.

Flash flood disasters are influenced by rainfall intensity, basin topography, physiognomy, soil types, land use types, and also by the position, density and vulnerability of building and population. These influential factors can be divided into 3 types: disaster-inducing factors, disaster-breeding environment and hazard-bearing bodies. Disaster-inducing factors refer to rainfall distribution, rainfall intensity, etc. Disaster-breeding environment mainly refer to basin’s underlying surface conditions impacting runoff production and flow routing. Hazard-bearing bodies mainly refer to the distribution, density and vulnerability of buildings, houses, etc.

In hilly regions, such as Licheng county in China which was chosen as an example of this paper, there may exist many small basins that have same disaster-inducing factors and hazard-bearing bodies, but with different disaster-breeding environment. Flash flood risk in this region is mainly affected by different disaster-breeding environment. So, in this paper, flash flood risk of small basins was evaluated by mainly considering basin’s underlying surface conditions.

There are many flood risk assessment methods which include hydrology and hydraulics models [4, 5], dynamic critical rainfall method [6, 7], artificial neural network [8, 9], fuzzy synthetic evaluation [10, 11], GIS&RS risk analysis evaluation methods [12, 13], variable fuzzy method [14 –17] and so on. Among these methods, hydrology and hydraulics models, GIS&RS risk analysis evaluation methods are usually employed to create guiding map for flood hazard evaluation [18, 19]. The dynamic critical rainfall method can calculate the rainfalls corresponding to different magnitude, and it is usually applied to mountain flood warning. Artificial neural network needs many samples, and its advantages and disadvantages are very different with the differences of network structures. For fuzzy synthetic evaluation, the constructions of membership function and weight vary with each individual, and it is difficult to generalize the evaluation model [20, 21]. Because basin’s underlying surface conditions are complex, the impact of each underlying surface factor on flash flood risk may have a relationship with randomness and fuzziness. Therefore, variable fuzzy method (VFM) is chosen to evaluate flash flood risk in this study. VFM, developed by Chen Shouyu in 2005 [20], is effective for fuzzy system with linear or nonlinear relationship. VFM can handle the fuzziness of hierarchical boundaries scientifically and reasonably [22, 23], can properly determines the comprehensive assessment grade of samples by varying the indexes of model, and this method is also flexible [24, 25], and is widely used in water resources evaluation [21, 26], water environment evaluation [27], water quality assessment [28, 29], flood control operations [30], land suitability evaluation [31], flood risk assessment [14 –17] and so on.

There are many studies on flood risk assessment by using variable fuzzy method. In literature [14] and [15], the authors establish the flood risk assessment index system of Jingjiang flood diversion district based on hazard and vulnerability. In literature [16] and [17], the authors evaluate disaster risk of China with 4 indexes of disaster area, inundated area, dead population and collapsed houses. Literature [14] only uses variable fuzzy sets theory to evaluate the flood hazard grade and flood vulnerability grade, respectively. Literature [15] combines set pair analysis and variable fuzzy sets theory in order to make the relative membership degree function calculation of variable fuzzy method simple. In literature [16] and [17], the methods used are variable fuzzy sets theory and information diffusion. The disaster degree values are calculated by variable fuzzy sets theory. On the basis of the disaster degree values, information diffusion is used to calculate probability of disaster degree values.

Although there are many studies on flood risk assessment by using variable fuzzy method, there are few flash flood risk assessments of small basin in flood-prone hilly regions. So in this paper, according to the characteristics of flash flood of small basin in flood-prone hilly regions, the flash flood risk index assessment system is established based on disaster-breeding environment. The flash flood risk is evaluated by using variable fuzzy method. In order to verify the result of VFM, fuzzy clustering analysis [FCA] is also used. FCA is a kind of multivariate analysis method for classification. FCA has been found extensive applications in many fields. In the field of water resources and environment, it has been used for water quality assessment [32], water resources carrying capacity evaluation [33], environmental quality evaluation and analysis [34] and stream flow time series clustering [35, 36]. In the field of agriculture, it has been used for agricultural soil resources evaluation and classification [37], regional land main-function division [38], soil nutrient classification [39]. In other fields, it is also widely used [40 –42].

The purpose of this study is to determine the priorities of numerous small basins in hilly regions based on VFM by mainly considering basin’s underlying conditions. This assessment system provides an important decision basis for effective mountain flash flood management under limited funds.

2. Methods

2.1. Variable fuzzy method

Based on the concept of fuzzy sets depicting imprecision or vagueness of Zadeh, Chen Shouyu established a fuzzy set theory and a engineering fuzzy set theory [22], and applied the theories to identify river water quality [23].Then the variable fuzzy method (VFM) or variable fuzzy sets theory was presented by Chen Shouyu which can scientifically and reasonably determine membership degrees and relative membership functions of objectives or indexes at level interval relating to engineering system [24, 25].

2.1.1 The definition of variable fuzzy sets

In order to define the concept of Variable Fuzzy Sets, we suppose that U is a fuzzy concept, A and A^c represent attractability and repellency, respectively. To any elements u (u ∈ U), μ_A (u) and μ_{A
^c} (u) are the relative membership function of element u to A and A^c that express degrees of attractability and repellency, respectively, where, μ_A (u) + μ_{A
^c} (u) =1, and 0 ≤ μ_A (u) ≤1, 0 ≤ μ_{A
^c} (u) ≤1 [22 , 25].

If D_A (u) is defined as relative difference degree of u to A, expressed as Equation (1) [22 , 25]. $D_{A} (u) = μ_{A} (u) - μ_{A^{c}} (u)$ (1)

Then, mapping ${\begin{array}{l} D_{A} : D \to [- 1, 1] \\ u | \to D_{A} (u) \in [- 1, 1] \end{array}$ (2)

Since μ_A (u) + μ_{A
^c} (u) =1 then $D_{A} (u) = 2 μ_{A} (u) - 1$ (3)

or $μ_{A} (u) = (1 + D_{A} (u)) / 2$ (4)

$\begin{matrix} V = {(u, D) | u \in U, \\ D_{A} (u) = μ_{A} (u) - μ_{A^{c}} (u), D \in [- 1, 1]} \end{matrix}$ (5) $A_{+} = {u | u \in U, 0 < D_{A} (u) \leq 1}$ (6) $A_{-} = {u | u \in U, - 1 \leq D_{A} (u) < 0}$ (7) $A_{0} = {u | u \in U, D_{A} (u) = 0}$ (8) Where V is Variable Fuzzy Sets, A₊, A_- and A₀ indicate attracting sets, repelling sets and qualitative change boundary of Variable Fuzzy Sets, respectively [22].

Suppose X₀ ∈ [a, b] is attracting sets of variable fuzzy set in real number axis, i.e. 0 < D_A (u) ≤1. X ∈ [c, d] are the upper bound and lower bound of a certain range which includes X₀ (X₀ ⊂ X). According to the definition of variable fuzzy set V, it is known that [c, a] and [b, d] are all repelling sets, i.e. -1 ≤ D_A (u) <0. Suppose that M is a point value belonging to the interval [a, b] and it’s relative difference degree D_A (u) =1, then it’s relative membership degree μ_A (u) =1 [22 , 25].

2.1.2 The relative membership degree of indexes

Suppose there are n samples and each sample has m index values, then x_ij is the value of index i of the sample j; i = 1, 2, …, m; j = 1, 2, …, n.

Suppose the samples are comprehensively evaluated by l grading levels. According to the values of each index for different grade levels, the matrix of attraction range I_ab, the matrix of range I_cd and the point value matrix M can be computed as follows: $I_{ab} = [\begin{matrix} [a_{11}, b_{11}] & [a_{12}, b_{12}] & \dots & [a_{1 c}, b_{1 l}] \\ [a_{21}, b_{21}] & [a_{22}, b_{22}] & \dots & [a_{2 c}, b_{2 l}] \\ ⋮ & ⋮ & ⋮ & ⋮ \\ [a_{m 1}, b_{m 1}] & [a_{m 2,} b_{m 2}] & \dots & [a_{mc}, b_{ml}] \end{matrix}]$ (9)

Where [a_ih, b_ih] is attraction scope of each index for different grade levels; i = 1, 2, …, m; h = 1, 2, …, l.

Level 1 means the superior level and level l means the inferior level. If indexes belong to the type of the larger the better, then a > b; if indexes belong to the type of the less the better, then a < b. For every [a_ih, b_ih], the range of interval [c_ih, d_ih] can be determined according to the upper and lower bound of its adjacent intervals. $I_{cd} = [\begin{matrix} [c_{11}, d_{11}] & [c_{12}, d_{12}] & \dots & [c_{1 c}, d_{1 l}] \\ [c_{21}, d_{21}] & [c_{22}, d_{22}] & \dots & [c_{2 c}, dl] \\ ⋮ & ⋮ & ⋮ & ⋮ \\ [c_{m 1}, d_{m 1}] & [c_{m 2,} d_{m 2}] & \dots & [c_{mc}, d_{ml}] \end{matrix}]$ (10)

Where [c_ih, d_ih] is the range of each index for different grade levels; i = 1, 2, …, m; h = 1, 2, …, l. M_ih is an important index and can be obtained according to a_ih and b_ih as follows: $M_{ih} = \frac{l - h}{l - 1} a_{ih} + \frac{h - 1}{l - 1} b_{ih}$ (11)

Where M_ih is a point belonging to [a_ih, b_ih], and its relative membership degree equals to 1. i = 1, 2, …, m; h = 1, 2, …, l.

Equation (11) satisfies the following three special conditions: (1) when h = 1, then M_i1 = a_i1; (2) when h = l, then M_il = b_il; (3) when $h = \frac{l + 1}{2}$ then $M_{ih} = \frac{a_{ih} + b_{ih}}{2}$ .

If x_ij belongs to the interval [a_ih,b_ih], and x_ij is located at the left of M_ih, the relative membership degree μ_A (x_ij) _h of x_ij to level h can be calculated as follows [22 , 25]: $\begin{matrix} μ_{A} (x_{ij})_{h} \\ = {\begin{matrix} 0.5 (1 + \frac{x_{ij} - a_{ih}}{M_{ih} - a_{ih}}) & x_{ij} \in [a_{ih}, M_{ih}] \\ 0.5 (1 - \frac{x_{ij} - a_{ih}}{c_{ih} - a_{ih}}) & x_{ij} \in [c_{ih}, a_{ih}] \end{matrix} \end{matrix}$ (12)

On the other hand, if x_ij belongs to the interval [a_ih,b_ih], and x_ij is located at the right of M_ih, the relative membership degree of x_ij to level h can be calculated as follows [22 , 25]:

$\begin{matrix} μ_{A} (x_{ij})_{h} \\ = {\begin{matrix} 0.5 (1 + \frac{x_{ij} - b_{ih}}{M_{ih} - b_{ih}}) & x_{ij} \in [M_{ih}, b_{ih}] \\ 0.5 (1 - \frac{x_{ij} - b_{ih}}{d_{ih} - b_{ih}}) & x_{ij} \in [b_{ih}, d_{ih}] \end{matrix} \end{matrix}$ (13) $μ_{A} (x_{ij})_{h} = 0 x_{ij} \notin [c_{ih}, d_{ih}]$ (14)

Finally, if x_ij does belong to the interval [c_ih,d_ih], then D_A (u) = −1, μ_A (x_ij) _h = 0.

Equations (12 and 13) satisfy the following special conditions: (1) when x_ij = a_ih or x_ij = b_ih, then D_A (u) =0, μ_A (x_ij) _h = 0.5; (2) when x_ij = M_ih, then D_A (u) =1, μ_A (x_ij) _h = 1; (3) when x_ij = c_ih or x_ij = d_ih then D_A (u) = -1, μ_A (x_ij) _h = 0.

The matrix of the relative membership degree, U_j, can be computed by Equations (12–14): $U_{j} = [μ_{A} (x_{ij})_{h}]$ (15)

Further, a synthetic relative membership degree vector, ${ju}_{h}^{'}$ , be obtained by the variable fuzzy set theory [22 , 25]. ${ju}_{h}^{'} = {1 + {[\frac{\sum_{i = 1}^{m} [ω_{i} (1 - μ_{A} (x_{ij})_{h})]^{β}}{\sum_{i = 1}^{m} [ω_{i} μ_{A} (x_{ij})_{h}]^{β}}]}^{α / β}}^{- 1}$ (16)

Where ω is the weight of each index. The weight of each index represents its contribution to flash flood risk, and it has a profound effect on the assessment results. α is a rule index of optimization model (α = 1 for least single method and α = 2 for least square method); and β is distance index of optimization model (β = 1 denotes hamming distance and β = 2 denotes Euclidean distance). There are four kinds of combinations for different α and β.

According to Equation (16), the synthetic relative membership degree of each index about the grade level h is obtained: $U_{j}^{'} = [_{j} u_{h}^{'}]$ (17)

After normalizing ${ju}_{h}^{'}$ , the normalized synthetic relative membership degree of each index about the grade level h is obtained: $U_{j}^{″} = [_{j} u_{h}]$ (18)

${Where}_{j} u_{h} = \frac{_{j} u_{h}^{'}}{{\sum_{h = 1}^{l}}_{j} u_{h}^{'}}$

At last, the grading level of each sample can be computed by Equaiton (19): $H_{j} = [1 2 \dots l] U_{j}^{'}$ (19)

Compute the level eigenvalues H, then, the final evaluation level can be determined in accordance with judging criteria, which are described as follows:

If 1.0 ≤ H ≤ 1.5, then the evaluation result is level one; else if h - 0.5 < H ≤ h, then the evaluation result is level h–1; else if h < H ≤ h + 0.5, then the evaluation result is level h + 1; else if l - 0.5 < H ≤ l, then the evaluation result is level l. In this paper, l = 3. If 1.0 ≤ H ≤ 1.5, then the evaluation result is level one; else if 1.5 < H ≤ 2.5, then the evaluation result is level two; else if 2.5 < H ≤ 3.0 then the evaluation result is level three.

2.1.3 Weight calculation by analytic hierarchy process

To determine the weight of each index is an important part in VFM. Because the weight of each index represents its contribution to the risk of disaster-breeding environment, it has a profound effect on results. Currently, the Analytic Hierarchy Process (AHP) weighting method is widely used in combination with the qualitative and quantitative methods [43 –45]. The main calculation steps of AHP are as follows:

Construct a judgment matrix. The judgment matrix reflects the relative importance of the correlation factors between the factor in upper layer and the multiple factors in lower layer. According to the relative importance of each factor which determined based on the hierarchical structure of index system and the mutual influence of all elements, the relative importance of each factor based on 1–9 scales is defined [43], then the judgment matrix can be constructed.

Calculate the maximum eigenvalue and eigenvector of the judgment matrix with the square root method or the geometric average method.

Check the consistence of the judgment matrix. To ensure a reasonable judgment matrix, the judgment matrix’s consistency needs to be checked as following. Suppose CR is the ratio between the consistency index and random index, “CR < 0.1” will indicate that the judgment matrix has good transitivity and consistency [43].

CI = \frac{λ_{{max}^{- n}}}{n - 1}

(20)

CR = \frac{CI}{RI}

(21)

Where CR is consistency ratio; CI is the consistency index; λ_max is the maximum eigenvalue of the judgment matrix; n is the order of the judgment matrix; RI is random index.

(4) Calculate the indexes weights. According the weight of index layer relative to criteria layer, and the weight of criteria layer relative to objective layer, the weight of each index layer relative to objective layer can be calculated. The detailed calculation steps can be found in literatures [43, 45].

2.2. Fuzzy clustering analysis

Fuzzy clustering analysis [FCA] is a method which makes use of fuzzy mathematics to ascertain the closeness degree of samples. Supposed a fuzzy subset R = (r_ij) _n×n, r_ij ∈ [0, 1], which belong to a given finite set X. If the fuzzy subset satisfies (1) r_ii = 1; (2) r_ij = r_ji (i, j = 1, 2, …, n); and (3)R ∘ R ⊆ R; then a fuzzy equivalence matrix of R = (r_ij) _n×n can be structured. Selecting a corresponding confidence level λ, samples can be classified [34, 35]. The main steps are as follows:

Data standardization. In practice, the dimensions and magnitudes of original data are different. In order to eliminate the influences of king-sized values, a dimensionless processing will be executed at first.

Fuzzy similar matrix building. By computing the similar degree r_ij between samples x_i and x_j, a fuzzy similar matrix R = (r_ij) _n×n will be built. In this paper, Euclidean distance method is used to calculate r_ij [34, 35].

Fuzzy equivalence matrix building. Using transitive closure t (R), fuzzy equivalence matrix t (R) = (r_ij) is built. According to square method, it is R → R² → R⁴ → … → R^2k, R^2k = R^2(k+1) can be obtained when k is big enough, and then let t (R) = R^2k [34, 35].

Cluster analysis. Given a confidence level λ, matrix [t (R) _λ] can be computed. Usually, for a given fuzzy equivalent matrix t (R) = (r_ij), ∀λ ∈ [0, 1], when r_ij ⩾ λ, we can get λr_ij = 1; when r_ij < λ, we can get λr_ij = 0. Then let λ, fall down from 1 to 0, [t (R) _λ] is ascertained. At last, x_i and x_j are converged into one category on the condition of R₂ (x_i, x_j) =1 (i, j = 1, 2, …, n). The detailed calculation steps can be found in literatures [34, 35].

3. Case study

Licheng County, located at the transition zone between mountainous areas in the middle of Shandong province, China, has a area 1298km² with E116°49′ ∼ 117°29′ and N36°20′ ∼ 36°62. As shown in Fig. 1. Licheng County belongs to warm-temperate zone, monsoon climate with four different seasons. With an uneven precipitation distribution in space, the average annual precipitation is 665.7 mm, which takes place mainly in 6∼9 month accounting for 66% of the annual rainfall. Licheng county is prone to flash flood and the one of the pilot counties of flash flood mitigation in China [46]. In recent years, flash floods often lead to serious disasters. Big flash flood disasters happened in the year of 1963, 1964,1985,1987,1994, 1996, 2000 and 2010, with seriously destroyed bridges and roads, inundated farmland and buildings, and a direct economic loss of about 99.1 million RMB [47, 48]. Flash flood disaster has become a prominent problem and a restraining factor to Licheng’s sustainable development. According to the survey of the flash flood disaster, there are 119 small basins listed as flash flood disaster prevention in Licheng county shown in Fig. 1. In order to manage flash flood disaster properly, risk evaluation of disaster-breeding environment was carried off here.

Fig. 1

Location and small basin distribution in Licheng County (a) Location of Licheng county in China (b) Small basins distribution in Licheng county.

3.1. Index selection

Flash flood is caused by heavy rain, the underlying surface conditions. In small hilly basins, because of small basin area, low stream density, steeper basin slope, steeper channel slopes and short flow routing time, flash floods are characterized by fast flood rise and drop, large peak flow, and small runoff volume. Peak flow is the main consideration in determining the scale of water conservancy projects and flash flood warning projects. For same rainfall conditions, flash flood is mainly affected by basin’s underlying surface conditions. In this paper, small basin’s underlying surface conditions affecting flash flood are considered only. Focusing on the impacts on the flood peak, the evaluation index system including objective layer, criterion layer and index layer has constructed. The index system has 9 indexes which are selected according to local conditions and reference literatures [18, 19]. The objective layer represents risk level of disaster-breeding environment of small basin. The criterion layer is the factors of runoff production and flow routing. The index layer is the main factors influencing runoff production and flow routing of disaster-breeding environment of small basin. The main factors influencing runoff production are basin area (F) and soil moisture (CN). The main factors influencing flow routing are basin shape factor (K_e), basin slope (S), longest flow path length(L₁), slope of the longest flow path(J₁), main stream length (L₂), slope of main stream (L₂) and basin centroid height(H₀).

According to the evaluation indexes, the characteristic value of each index of each small basin in Licheng county is shown in Table 1.

Table 1
Characteristic values of each index of each small basin in Licheng county

Code of basin	F(km²)	Ke	S	CN	L₁ (km)	J₁	L₂ (km)	J₂	H₀ (m)
WDA8500121F00000	9.522	0.33	0.379	68	5.402	0.039	2.382	0.019	193.94
WDA8500125000000	16.111	0.33	0.464	68	6.983	0.031	3.488	0.011	482.9
WDD1100121C00000	16.233	0.22	0.365	67	8.569	0.022	0.0001	0.00001	314.37
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
WDA850012B000000	17.391	0.34	0.201	83	7.181	0.008	10.897	0.025	139.44
WDD1100121K00000	31.292	0.2	0.013	86	12.65	0.003	0.135	0.001	25.06
WDD1100128000000	4.221	0.22	0.009	78	4.403	0.0002	3.974	0.0004	25.43

3.2. Rank division for evaluation

Owing to the different nature of each index, some indicators, such as CN, basin slope, positively correlate with flash flood risk. However, some indicators such as longest flow path length negatively correlate with flash flood risk. In order to facilitate the risk evaluation of disaster-inducing environment, statistical analysis is carried on the following characteristic values of standard deviation, mean, median, the maximum and minimum values of each index. Based on this statistical analysis and reference literatures [14 –17], 3 risk levels are divided and they are high, medium and low. The ranges of each index for different risk levels are determined and shown in Table 2.

Table 2
Ranges of each index for different risk levels

Risk level l F (km²) Ke S CN L₁(km) J₁ L₂(km) J₂ H₀(m)

High 1 30–50 0.29–0.58 0.359–0.645 70–90 0.096–8.52 0.045-0.107 0–6.5 0.019–0.046 350–700

Medium 2 20–30 0.15–0.29 0.139–0.359 50–70 8.52–17.6 0.015-0.045 6.5–19 0.0065–0.019 150–350

Low 3 0–20 0.01–0.15 0.0002–0.139 30–50 17.6–31.9 0-0.015 19–26 0–0.0065 0–150

Risk level	l	F (km²)	Ke	S	CN	L₁(km)	J₁	L₂(km)	J₂	H₀(m)
High	1	30–50	0.29–0.58	0.359–0.645	70–90	0.096–8.52	0.045-0.107	0–6.5	0.019–0.046	350–700
Medium	2	20–30	0.15–0.29	0.139–0.359	50–70	8.52–17.6	0.015-0.045	6.5–19	0.0065–0.019	150–350
Low	3	0–20	0.01–0.15	0.0002–0.139	30–50	17.6–31.9	0-0.015	19–26	0–0.0065	0–150

Note: l— grading level.

3.3. Evaluation result

According to Table 1, the characteristic value matrix X of indices of 119 small basins is computed. $X = [\begin{matrix} 9.522 & 16.111 & \dots & 31.292 & 4.221 \\ 0.33 & 0.33 & \dots & 0.20 & 0.22 \\ 0.379 & 0.464 & \dots & 0.013 & 0.009 \\ 68 & 68 & \dots & 86 & 78 \\ 5.402 & 6.983 & \dots & 12.65 & 4.403 \\ 0.039 & 0.031 & \dots & 0.003 & 0.0002 \\ 2.382 & 3.488 & \dots & 0.135 & 3.974 \\ 0.019 & 0.011 & \dots & 0.001 & 0.0004 \\ 193.94 & 482.9 & \dots & 25.06 & 25.43 \end{matrix}]$

Based on Table 2 and Equation (9), the criteria interval matrix I_ab can be computed. $I_{ab} = [\begin{matrix} [50, 30] & [30, 20] & [20, 0] \\ [0.58, 0.29] & [0.29, 0.15] & [0.15, 0.01] \\ [0.645, 0.359] & [0.359, 0.139] & [0.139, 0.0] \\ [90, 70] & [70, 50] & [50, 30] \\ [0.096, 8.52] & [8.52, 17.6] & [17.6, 31.9] \\ [0.107, 0.045] & [0.045, 0.015] & [0.015, 0] \\ [0.0, 6.5] & [6.5, 19] & [19, 26] \\ [0.046, 0.019] & [0.019, 0.0065] & [0.0065, 0.0] \\ [700, 350] & [350, 150] & [150, 0] \end{matrix}]$

According to the criteria interval matrix I_ab and Equation (10), the matrix I_cd can be obtained. The M_ih can be obtained according to Equation (11) and the standard eigenvalue a_ih and b_ih in matrix I_ab. $I_{cd} = [\begin{matrix} [50, 20] & [50, 0] & [30, 0] \\ [0.58, 0.15] & [0.58, 0.01] & [0.29, 0.01] \\ [0.645, 0.139] & [0.645, 0.0] & [0.359, 0.0] \\ [90, 50] & [90, 30] & [70, 30] \\ [0.096, 17.6] & [0.096, 31.9] & [8.52, 31.9] \\ [0.107, 0.015] & [0.107, 0] & [0.045, 0] \\ [0.0, 19] & [0.0, 26] & [6.5, 26] \\ [0.046, 0.0065] & [0.046, 0.0] & [0.019, 0.0] \\ [700, 150] & [700, 0] & [350, 0] \end{matrix}]$ $M = [\begin{matrix} 50 & 25 & 0 \\ 0.58 & 0.295 & 0.01 \\ 0.645 & 0.323 & 0.0 \\ 90 & 60 & 30 \\ 0.096 & 16.0 & 31.9 \\ 0.107 & 0.054 & 0.0 \\ 0.0 & 13 & 26 \\ 0.046 & 0.023 & 0.0 \\ 700 & 350 & 0 \end{matrix}]$

Taking the first basin (WDA8500121F00000) as shown in Table 1) as an example, by using Equations (12–14), the relative membership degree matrix U₁ is computed. $U_{1} = [μ_{A} (x_{i 1})_{h}] = [\begin{matrix} 0 & 0.238 & 0.762 \\ 0.569 & 0.431 & 0 \\ 0.534 & 0.466 & 0 \\ 0.450 & 0.600 & 0.050 \\ 0.685 & 0.315 & 0 \\ 0.402 & 0.809 & 0.098 \\ 0.817 & 0.183 & 0 \\ 0.484 & 0.867 & 0.016 \\ 0.110 & 0.610 & 0.390 \end{matrix}]$

Based on analytic hierarchy process (AHP) method discussed in section 2.1.3, the weight of each indicator is w = (0.367, 0.086, 0.134, 0.183, 0.033, 0.079, 0.029, 0.061, 0.028).

When α = 1, β = 2, by Equation (16), the synthetic relative membership degree matrix $U_{1}^{'}$ of each index about the level h is obtained. $U_{1}^{'} = [_{1} u_{h}^{'}] = [\begin{matrix} 0.251 \\ 0.371 \\ 0.507 \end{matrix}]$

Normalizing $_{1} u_{h}^{'}$ , the normalized synthetic relative membership degree $_{1} u_{h}^{''}$ of each index about the level h is obtained. $U_{1}^{'} [\begin{matrix} 0.222 \\ 0.329 \\ 0.449 \end{matrix}]$

By Equation (19), the characteristic value of risk level of small basin (WDA8500121F00000) is obtained. $H = [\begin{matrix} 1 & 2 & 3 \end{matrix}] [\begin{matrix} 0.222 \\ 0.329 \\ 0.449 \end{matrix}] = 2.227$

In a similar way, when α = 1, β = 1; α = 2, β = 1 and α = 2, β = 2, the characteristic values of risk level of small basin (WDA8500121F00000) are obtained in Table 3. Then the average of risk level based on different α and β are calculated and shown in Table 3.

Table 3
Characteristic value of risk level of small basin (WDA8500121F00000)

H				Average value of H
β = 1	β = 1	β = 2	β = 2
α = 1	α = 2	α = 1	α = 2
1.995	1.991	2.227	2.473	2.172

Finally, the characteristic values of risk level of 119 small basins are computed and shown in Table 4. According to the evaluation results, combining judging criteria given in section 2.1.2, the risk levels of disaster-breeding environment for 119 small basins are determined and shown in Table 4.

Table 4

Characteristic values of risk level of 119 small basins in Licheng county

Code of basin	H				Average value of H	Risk level
	β = 1	β = 1	β = 2	β = 2
	α = 1	α = 2	α = 1	α = 2
WDA8500121F00000	1.995	1.991	2.227	2.473	2.172	2
WDA8500125000000	1.957	1.925	2.150	2.294	2.081	2
WDD1100121C00000	2.108	2.176	2.207	2.367	2.215	2
WDD1100122I00000	2.333	2.679	2.182	2.405	2.400	2
⋮	⋮	⋮	⋮	⋮	⋮	⋮
WDA8500122E00000	2.171	2.251	2.211	2.384	2.254	2
WDA850012B000000	2.046	2.093	2.126	2.270	2.134	2
WDD1100121K00000	1.907	1.767	1.807	1.585	1.766	2
WDD1100128000000	2.439	2.818	2.370	2.734	2.590	3

Table 4 shows that there is no basin belonging to high risk level, 14 basins belonging to low risk level, and 105 small basins belong to medium risk level. Risk level distribution of 119 small bas ins is mapped out by using Arcgis software and shown in Fig. 2. In order to identify risk difference of small basins for the same risk level, based on the average value of H, the risk priorities of 119 small basins are sorted and mapped out by using Arcgis software in Fig. 3. This map will provide a great support to risk management of the 119 small basins.

Fig. 2

Risk levels of 119 small basins.

Fig. 3

Risk priorities diagram of 119 small basins based on average value of H.

3.4. Fuzzy clustering analysis

Using standard deviation method, the standardized data from original data of 9 indexes for 119 small basins are computed. Euclidean distance method and transitive closure t (R) are used to build fuzzy similar matrix and fuzzy equivalent matrix, respectively. For a given confidence level λ, the fuzzy equivalent matrix is clustered. The result of fuzzy clustering for 119 small basins is shown in Fig. 4.

Fig. 4

Fuzzy clustering analysis of 119 small basins.

According to the clustering result as shown in Fig. 4, 3 classes are divided: Class 1 with 0.8351 < λ ≤ 0.970; Class 2 with 0.9700 < λ ≤ 0.9985; and Class 3 with 0.9985 < λ ≤ 1.0. In Class 1, there are 6 basins which are WDA8500124000000, WDA8500124G00000, WD A0000801A00000, WDA850012B000000, WDA8500 12E000000 and WDA8500121A00000. In Class 3, there are 12 basins which are WDD110012A000000, WDD 110012D000000, WDD110012B000000, WDD110012F000000, WDD110012C000000, WDD110012E000000, WDD1100122J00000, WDD1100127000000, WDD110012G000000, WDD1100122I00000, WDD11402BB000000 and WDD11402C0000000. The rema-ining 101 small basins belong to class 2.

3.5. Discussion

3.5.1 Basin characteristics of risk levels

From Fig. 2, it can be seen that the 14 small basins of risk level 3 are all located in the north of Licheng county, with several basin characteristics as follows: basin area is less than 12km²; basin shape factor is between 0.02 and 0.22; basin slope is less than 0.0086; slope of the longest flow path is less than 2.3%; basin centroid height is between 20.95m and 28.51m. These basin characteristics are shown in Table 5. From the average value of H in Fig. 3, it can be found that most basins of risk level 2 are also located in the north of licheng county with low average value H. There is a statistics of 23 small basins of risk level 2 whose H are less than 2. This statistics shows the following basin characteristics: basin area is more than 21km²; basin shape factor is between 0.06 and 0.29; basin slope is less than 0.443; slope of the longest flow path is less than 29.5%; basin centroid height is higher. These basin characteristics are show in Table 6.

Table 5
Basin characteristics of risk level 3

Code of basin	F(km²)	Ke	S	CN	L₁(km)	J₁	L₂(km)	J₂	H₀(m)	Average value of H	Risk level
WDD110012C000000	0.723	0.04	0.0007	78	4.195	0.0001	3.989	0.0001	21.73	2.668	3
WDD1100127000000	0.781	0.14	0.0071	75	2.329	0.0023	1.906	0.0019	28.51	2.655	3
WDD110012A000000	0.428	0.04	0.0011	80	3.337	0.0009	1.875	0.0007	22.93	2.652	3
WDD110012F000000	1.648	0.07	0.0002	80	4.95	0.0001	3.012	1E-05	20.95	2.647	3
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
WDD110012H000000	11.321	0.02	0.0006	81	23.39	0.0001	18.95	0.0001	21.16	2.601	3
WDD110012G000000	0.204	0.17	1E-05	84	1.105	0.00001	0.995	1E-05	20.95	2.597	3
WDD1100128000000	4.221	0.22	0.0086	78	4.403	0.0002	3.974	0.0004	25.43	2.59	3
WDD1100122P00000	9.54	0.1	0.0016	85	9.622	0.0008	6.262	0.0004	21.17	2.552	3

Table 6

Basin characteristics of average value of H less than 2 in risk level 2

Code of basin	Fs(km²)	Ke	S	CN	L₁(km)	J₁	L₂(km)	J₂	H₀(m)	Average value of H	Risk level
WDA8500123G00000	22.55	0.18	0.340	74	11.206	0.0182	6.71	0.0098	300.49	1.999	2
WDA8500129000000	22.961	0.21	0.427	70	10.402	0.0115	8.186	0.0069	196.87	1.994	2
WDD1100121L00000	21.004	0.28	0.385	74	8.738	0.0201	6.105	0.0164	280.24	1.983	2
WDA8500123000000	22.071	0.28	0.395	76	8.835	0.0207	5.105	0.0123	603.23	1.922	2
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
WDA8500122GB0000	28.833	0.29	0.367	76	9.993	0.0244	5.065	0.0133	247.57	1.631	2
WDD1100121M00000	41.705	0.23	0.003	84	13.395	1E-05	0.447	0.00001	21.38	1.621	2
WDD1100121U00000	41.8	0.28	0.001	83	12.259	0.0001	0.0001	0.00001	20.62	1.6	2
WDD1100121P00000	42.163	0.06	0.176	82	27.178	0.0086	23.9	0.0073	169.67	1.573	2

The above basin characteristics show that the greater basin area, basin shape factor, basin slope and slope of the longest flow path, the higher risk of basins.

3.5.2 Result comparison between VFM and FAC

The results from fuzzy clustering analysis (FAC) and variable fuzzy method (VFM) are compared here. According to VFM, there are 6 basins of class 1 based on FAC belong to risk level 2 of VFM. The average value of H of 6 basins is mainly concentrated around 2.1 shown in Table 7.

Table 7
Result comparison between VFM and FAC (for basins belong to class1)

FAC	VFM
Basins belong to class 1	Average value of H	Risk level
WDA8500124000000	2.185	2
WDA8500124G00000	2.413	2
WDA0000801A00000	2.168	2
WDA850012B000000	2.134	2
WDA850012E000000	1.794	2
WDA8500121A00000	2.188	2

Based on FAC, for the 12 small basins of class 3, 9 basins belong to risk level 3, and 3 basins belong to risk level 2. The average value of H of 12 basins is mainly concentrated around 2.6 shown in Table 8. For the remaining 101 small basins of class 2, there are 96 basins belong to risk level 2, and 5 basins belong to risk level 3. The comparison shows that the results from FAC and VFM are nearly in consistence.

Table 8

Result comparison between VFM and FAC (for basins belong to class3)

FAC	VFM
Basins belong to class 3	Average value of H	Risk level
WDD110012A000000	2.652	3
WDD110012D000000	2.626	3
WDD110012B000000	2.642	3
WDD110012F000000	2.647	3
WDD110012C000000	2.668	3
WDD110012E000000	2.639	3
WDD1100122J00000	2.644	3
WDD1100127000000	2.655	3
WDD110012G000000	2.597	3
WDD1100122I00000	2.4	2
WDD11402BB000000	2.358	2
WDD11402C0000000	2.229	2

4. Conclusions

In hilly regions, according to the characteristics of mountain flash flood in small basins, the evaluation index system of mountain flash flood risk has been built with 9 indexes based on disaster-breeding environment.

Licheng county is one of flash flood-prone regions in China. The characteristic values of 9 indexes of 119 small basins in licheng county are collected. According to evaluation criteria, there are 14 basins belonging to low risk level and 105 small basins belong to medium risk level by VFM. From the evaluation results of VFM, it can be seen that the basin area, basin shape factor, basin slope and slope of the longest flow path are main factors. FAC is used to further verify the reliability and rationality of VFM results. The evaluation result comparison between VFM and FAC shows that they are nearly in consistence.

The small basins with low risk level are all located in the north of Licheng county as shown in Fig. 2. According to the statistics based on long historical observation data, the above results are consistent with Licheng’s reality. Therefore, the VFM can be used in flash flood risk evaluation of disaster-breeding environment.

For numerous small hilly basins that need flash flood prevention and control with limited funds, it is necessary to give a priority order or to determine which basin needs to be harnessed firstly. This paper, based on underlying surface conditions, establishes a mountain flash flood risk assessment system. Using this assessment system, the priority order of harness for many small basins is put forward by using VFM (as shown in Fig. 3). It provides an important decision basis for effective mountain flash flood management under limited funds.

At the same time, this paper points out several main factors affecting mountain flash flood risk. Based on these main influencing factors, the flash floods of small basins can be mitigated more effectively through a more comprehensive management. The efficacy of mountain flash flood risk management will be more prominent.

Footnotes

Acknowledgments

This work was supported by (1) the project “Flash Flood Disaster Prevention and Control Technology Application Research in Shandong Province” (Project No: SZJSYY-YJ201501, SZJSYY-YJ201502), and (2) the project “Shandong Provincial Water Conservancy Scientific Research and Technology Promotion” (Project No: SDSLKY201402).

Authors thank anonymous reviewers for their constructive and helpful feedback on this manuscript.

References

Špitalar ,

J.J.

Gourley ,

Lutoff ,

P.-E.

Kirstetter ,

Brilly and

Carr , Analysis of flash flood indexes and human impacts in the US from 2006 to 2012, Journal of Hydrology 519 (2014), 863–870.

J.D.

Creutin ,

Borga ,

Gruntfest ,

Lutoff and

Zoccatelli , A space and time framework for analyzing human anticipation of flash floods, Journal of Hydrology 482 (2013), 14–24.

Jin , The current situation, problems and countermeasures of mountain flash flood disaster control, China Water Resources 14 (2016), 9–11.

Vincendona ,

Ducrocq ,

G.-M.

Saulnier ,

Bouilloud ,

Chancibault ,

Habets and

Noilhan , Benefit of coupling the ISBA land surface model with a TOPMODEL hydrological model version dedicated to Mediterranean flash-floods, Journal of Hydrology 394 (2010), 256–266.

Chena ,

Yanga ,

Hong ,

J.J.

Gourley and

Zhang , Hydrological data assimilation with the Ensemble Square-Root-Filter: Use of streamflow observations to update model states for real-time flash flood forecasting, Advances in Water Resources 59 (2013), 209–220.

K.P.

Georgakakos , Analytical results for operational flash flood guidance, Journal of Hydrology 317 (2006), 81–103.

Hongping ,

Shu ,

Y.Y.

Liu and

H.U.

Chang-Wei , Research on the critical rainfall of flash floods in mountainous areas, Journal of China Institute of Water Resources and Hydropower Research 12 (2014), 185–189.

Shaofei ,

Ping and

Shuhong , Fuzzy risk assessment of flood hazard based on artificial neural network for detention basin, China Rural Water and Hydropower 6 (2008), 60–64.

Abdellatif ,

Atherton ,

Alkhaddar and

Osman , Flood risk assessment for urban water system in a changing climate using artificial neural network, Natural Hazards 79 (2015), 1–19.

10.

Minshen and

Chengchen , Research on grade model of flood risk assessment, Journal of Catastrophology 22 (2007), 1–5.

11.

Pei ,

Tairu and

Jun , GIS and hazy evaluation method based on hazard evaluation for flood disaster, taking chongqing as an example, Journal of China West Normal Univer8ity (Natural Sciences) 28 (2007), 165–171.

12.

Sanyal and

X.X.

Lu , GIS-based flood hazard mapping at different administrative scales: A case study in Gangetic West Bengal, India, Singapore Journal of Tropical Geography 27 (2006), 207–220.

13.

M.E.

Bastawesy ,

White and

Nasr , Integration of remote sensing and GIS for modelling flash floods in Wadi Hudain catchment, Egypt, Hydrological Processes 23 (2009), 1359–1368.

14.

Qiang ,

Jianzhon ,

Chao ,

Lixiang ,

Jun and

Xiaoling , Flood disaster risk analysis based on variable fuzzy sets theory, Transactions of the Chinese Society of Agricultural Engineering 28 (2012), 126–132.

15.

Zou ,

J.Z.

Zhou ,

L.X.

Song and

Guo , Comprehensive flood risk assessment based on set pair analysis-variable fuzzy sets model and fuzzy AHP, Stoch Env Res Risk A 27 (2013), 525–546.

16.

Li ,

Zhou ,

Liu and

Jiang , Research on flood risk analysis and evaluation method based on variable fuzzy sets and information diffusion, Safety Science 50 (2012), 1275–1283.

17.

Li ,

Zhou ,

Liu ,

Tang and

Zou , Disaster risk assessment based on variable fuzzy sets and improved information diffusion method, Human and Ecological Risk Assessment 19 (2013), 857–872.

18.

Shanshan ,

Xiangjun and

Z.D.Z.

Hong , GIS-based flood risk assessment in suburban areas: A case study of the Fangshan District, Beijing, Natural Hazards 87 (2017), 1525–1543.

19.

Na and

Jing , Analysis on characteristics of torrential flood disaster and discussion on method of flood hazard mapping, Water resources and Hydropower Engineering 42 (2011), 57–61.

20.

Shouyu , Theory and model of engineering variable fuzzy set—mathematical basis for fuzzy hydrology and water resources, Journal of Dalian University of Technology 45 (2005), 308–312.

21.

Shouyu and

Jimin , Variable fuzzy assessment method and its application in assessing water resources carrying capacity, Journal of Hydraulic Engineering 37 (2006), 264–271.

22.

S.Y.

Chen , The theory and application of engineering fuzzy sets, National Defence Industry Press, Beijing, 1998.

23.

Chang ,

H.W.

Chen and

S.K.

King , Identification of river water quality using the fuzzy synthetic evaluation approach, Journal of Environmental Management 63 (2001), 293–305.

24.

S.Y.

Chen and

Guo , Variable fuzzy sets and its application in comprehensive risk evaluation for flood-control engineering system, Fuzzy Optim Decis Ma 5 (2006), 153–162.

25.

S.Y.

Chen , Theory and model of variable fuzzy sets and its application, Dalian university of technology press, Dalian, 2009.

26.

Wan ,

Zhong and

E.K.

Appiah-Adjei , Variable sets and fuzzy pating interval for water allocation options assessment, Water Resources Management 28 (2014), 2833–2849.

27.

Li ,

Tan ,

Wei ,

Chen and

Feng , A new method of change point detection using variable fuzzy sets under environmental change, Water Resources Management 28 (2014), 5125–5138.

28.

Wang ,

Xu ,

Chau and

G.-J.

Lei , Assessment of river water quality based on theory of variable fuzzy sets and fuzzy binary comparison method, Water Resources Management 28 (2014), 4183–4200.

29.

Wang ,

Sheng ,

Wang ,

Ma ,

Wu and

Xu , Variable fuzzy set theory to assess water quality of the meiliang bay in taihu lake basin, Water Resources Management 28 (2014), 867–880.

30.

X.J.

Wang ,

R.H.

Zhao and

Y.W.

Hao , Flood control operations based on the theory of variable fuzzy sets, Water Resources Management 25 (2011), 777–792.

31.

Chen ,

Chai and

Su , Variable fuzzy sets methods and application on land suitability evaluation, Transactions of the CSAE 23 (2007), 95–97.

32.

Fanhui and

Jun , Application of fuzzy clustering in evaluating water quality of river system, Journal of Liaoning Technical University(Natural Science) 29 (2010), 982–984.

33.

Yansheng and

Xingfang , Application of fuzzy clustering iterative model to the evaluation of water resources carrying capacity, Journal of Shandong University (Engineering Science) 37 (2007), 100–105.

34.

Lanxing , Environment quality appraisal analysis based on fuzzy clustering analysis method with regard to 5 cities in southwestern area of China, University of Shanghai for Science and Technology 31 (2009), 302–305.

35.

M.C.

Jr ,

P.J.

Schweitzer and

T.W.

White , Problem decomposition and data reorganization by a clustering technique, Operation Research 20 (1972), 993–1009.

36.

Corduas , Clustering steam flow time series for regional classification, Journal of Hydrology 407 (2011), 73–80.

37.

Yanzhi , Application of fuzzy cluster analysis in farming land resources evaluation, Journal of Anhui Agri Sci 35 (2007), 7759–7760.

38.

Yali ,

Qun and

Kaisheng , Regional land main-function division based on fuzzy cluster analysis—a case study of Jiangsu Province, Bulletin of Soil and Water Conservatio 29 (2009), 127–130.

39.

Liying , Soil nutrient evaluation based on GIS and fuzzy clustering analysis, Journal of Anhui Agri Sci 38 (2010), 8595–8596.

40.

Yingrong ,

Jian and

Liping , Forecasting flood in the hanjiang river by using the method of fuzzy cluster, Journal of China Hydrology 26 (2006), 60–62.

41.

Yan , Application of fuzzy cluster analysis in coal mine gas emission prediction, Coal Technology 29 (2010), 82–85.

42.

Yong ,

Zhenglu ,

Niansheng and

Gengchao , Research on fuzzy clustering and its application, Science of Surveying and Mapping 35 (2010), 163–165.

43.

Neshat ,

Pradhan and

Dadras , Groundwater vulnerability assessment using an improved DRASTIC method in GIS, Resources Conservation & Recycling 86 (2014), 74–86.

44.

Song ,

Chen ,

Tian ,

Lv ,

Zhang and

Liu , The ecological vulnerability evaluation in southwestern mountain region of China based on GIS and AHP method, Proc Environ Sci 2 (2010), 465–475.

45.

Solomon and

Rehan , Risk-based environmental decision-making using fuzzy analytic hierarchy process (F-AHP), Stochastic Environ Res Risk Assess 21 (2006), 35–50.

46.

Sun ,

Zhang and

Cheng , Framework of national non-structural measures for flash flood Disaster prevention in China, Water 4 (2012), 272–282.

47.

Meng ,

Cao and

Zhao , Jinan flood control practice and exploration, China Water & Power Press, Beijing, 2008.

48.

Xue , Research on small Watershed classification and early-warning indicators of flash flood disasters in prevention areas, Shandong University, 2015.

Flash flood risk evaluation based on variable fuzzy method and fuzzy clustering analysis

Abstract

Keywords

1. Introduction

2. Methods

2.1. Variable fuzzy method

2.1.1 The definition of variable fuzzy sets

3. Case study

Table 1 Characteristic values of each index of each small basin in Licheng county

Table 3 Characteristic value of risk level of small basin (WDA8500121F00000)

3.5.1 Basin characteristics of risk levels

Table 5 Basin characteristics of risk level 3

Table 7 Result comparison between VFM and FAC (for basins belong to class1)

Footnotes

Acknowledgments

References

Table 1
Characteristic values of each index of each small basin in Licheng county

Table 3
Characteristic value of risk level of small basin (WDA8500121F00000)

Table 5
Basin characteristics of risk level 3

Table 7
Result comparison between VFM and FAC (for basins belong to class1)