Risk assessment of rainstorm and flood disasters in China between 2004 and 2009 based on gray fixed weight cluster analysis

Abstract

Rainstorm and flood disasters frequently occur in China, causing heavy losses for people’s lives and property and reducing the capability of sustainable development of the national and local economy. In this study, the risks of the rainstorm and flood disasters are assessed for the Chinese mainland, excluding Hong Kong, Macao, and Taiwan and also employ the historical data of seven indicators, including the affected area of crops, the affected population, the direct economic loss, and etc., from 2004 to 2009. Based on the large 1,302 historical sample data, the impact of rainstorm and flood disasters were analyzed using the methodology of gray fixed weight cluster analysis according to disaster losses, which were divided into the three gray classes of high, medium, and low. The regional differences of the risk assessment of the rainstorm and flood disasters are discussed, and the dynamical risk zoning map is conducted. The results show a consistent conclusion with the actual losses of rainstorm and flood disasters over each administrative district, which can provide more scientific evidence for the relevant departments of disaster prevention and mitigation.

1 Introduction

Rainstorm and flood disasters are serious natural disasters because they tend to happen suddenly and frequently while covering a wide area. In this century, catastrophic flood disasters all over the world have taken place nearly 40 times. Flood is also one of the main meteorological disasters in China, and about 50 % of the population and 70 % of property are located in flood-threatened areas (Gu et al. 2011). In recent decades, as the global climate continues to warm, all kinds of extreme weather and climate events have occurred frequently, with the frequency of flood occurrences and the losses increasing year by year. Taking 2004–2009 for an example, the area of crops affected by rainstorm and flood disasters was 118,884,000 hectares, the affected population was 625.526 million, and the direct economic losses amounted to 403.27 billion RMB yuan (Ren Min Bi), accounting for 28.2 % of all the periods of meteorological disasters economic loss and accounting for 0.27 % of the GDP. In recent years, rainstorm and flood disaster risk research in China has attracted extensive attention. In order to cope with the long plagued rain floods, reduce rainstorm floods brought by secondary disasters, protect ecosystems, ensure economic and social development, safeguard national security and stability, and strengthen disaster prevention, emergency response and post-disaster recovery efforts, it is urgent to improve the level of flood risk research and provide solid scientific and technological support for sustainable socio-economic development, which is of great significance for implementing effective flood disaster management, the capacity building of meteorological disaster prevention, and mitigation tasks and the sustainable development of the national economy.

In recent years, scholars have made great progress in the research of flood disaster risk assessment and regionalization. For example, Jiang et al. (2009) adopts fuzzy comprehensive assessment (FCA), simple fuzzy classification (SFC), and the fuzzy similarity method (FSM) to assess flood disaster risk in Kelantan, Malaysia. Validation data, such as the flooded area, paddy area, urban area, residential area, and refuges, were overlaid to validate and analyze the accuracy of flood disaster risk. Shao and Xiang (2009) analyzed the characteristics and classification of meteorological disasters in Hubei province, concluding that meteorological disasters in Hubei province have characteristics of great harmfulness, severe damage, and high frequency of occurrence. Zhang et al. (2000) studied the degree of risk of flood zoning. Wang et al. (2005) built a dynamic monitoring system of agro-meteorological disasters by using the 3S (Remote sensing, RS; Geography information systems, GIS; Global positioning systems, GPS) technology. GIS technology has a good advantage in early disaster warning, monitoring, and evaluation. Chinese scholars have widely applied the technology in studying the risks of floods and regionalization, and they have achieved good results. The representative researchers are Li (2005), Chen (2008), Yao (2000), Wan et al. (2007), Zhang et al. (2011), Gashaw and Legesse (2011), Ramlal and Baban (2008), Dewan et al. (2007), Liu and Liu (2001), Islam and Sado (2000), and Luo et al. (2007).

However, the GIS technology owns a relative single function with disadvantages in poor ease of development for the system structure and difficulties in ensuring the quality of data sources and capturing the accurate meteorological disaster information, which also brings more difficulties to flood disaster risk assessment. Scholars try to use the soft evaluation technique theory to assess the disaster risks, such as fuzzy techniques and gray evaluation techniques. For examples, Goro et al. (2013) estimated by integrating a physical-based approach as a total runoff integrating pathways (TRIP) model with Gumbel distribution metrics. The resulting equations are used to predict potential flood damage based on gridded Japanese data for independent variables. Li et al. (2011) used the method of attribute hierarchical model (AHM) to assess the meteorological disaster risk, which provided a more practical and scientific basis for meteorological disaster risk assessment and decision-making problems. Guo and Zha (2010) used AHP to analyze the Anhui province flood disaster losses and flood risk zoning, having a very positive effect on the assessment of disaster losses, prediction accuracy, and flood control. Li et al. (2012) put forward a composite method based on variable fuzzy sets and information diffusion method for disaster risk assessment. Kyung et al. (2013) used three multi-criteria decision-making (MCDM) techniques to quantitatively evaluate and compare 19 flood risk vulnerabilities of South Korea, including present conditions. Hochrainer and Mechler (2011) assess the rationale and applicability of such deliberations given the dynamic nature of vulnerability and risk and discuss conditions for conducting similar transactions for Asian megacity risks.

Over the years, due to the relatively small number of data samples, the relatively low reliability of the data quality, the high uncertainty of the decision-making system in the data information, gray clustering analysis has unique advantages. This method has been widely applied to the field of environmental quality assessment and the water quality analysis system. For example, Hu et al. (2012) applied the gray clustering model to evaluate the ambient air quality in Fuzhou city from 2004 to 2008, believing that improved gray clustering analysis evaluations are more objective and accurate. Zhu and Wang (2009) analyzed pollutant impact on indoor air quality, achieving gray evaluation in indoor air quality. Liu and Wang (2004) used gray clustering analysis to analyze the dump-leaching impact on groundwater quality. Lin et al. (2008) employed the gray decision-making method to evaluate green engineering and proposed a green ecological evaluation system for Taiwan. Hu et al. (2010) used the gray system to assess the quality of drinking water in Jiaozuo city, in which the method was simple and the operability was not bad and therefore achieved good results. Ip et al. (2009) applied the gray correlation degree to a water environmental quality assessment. Jiao and Ma (2010) comprehensively evaluated water in the Changjiang River based on gray-fuzzy clustering analysis. He et al. (2002) studied the application of the gray clustering decision in a comprehensive evaluation of water quality. Combining AHP and gray clustering analysis, Tang and Li (2011) created meteorological disaster post-assessment methods for highway traffic during flood seasons.

For the risk assessment system of flood loss in mainland China (excluding Hong Kong, Macao, and Taiwan), the indicators measured are relatively small: The risk assessment of disaster losses only includes a small number of measureable indicators, such as disaster losses, affected population, and affected areas; on the other hand, it often involves larger data samples and longer history disaster data (time series data). This paper tries to adopt the gray fixed weight cluster technique for risk analysis of disaster losses: national flood disaster loss risk assessment and zoning research involving 31 provinces, municipalities, and autonomous regions. The gray clustering techniques will help to explore the rainstorm and flood disasters losses based on a large-sample data classification problems.

We use disaster data from 2004 to 2009 of Yearbook of Meteorological Disasters in China and take 31 provinces, municipalities, and autonomous regions (except for Hong Kong, Macao, and Taiwan) as the clustering objects with seven disaster indicators, classifying the rainstorm flood losses of all 31 provinces, municipalities, autonomous regions (administrative divisions) by using a gray fixed weight cluster analysis. The trend of the future flood risk loss is predicted, which provides a scientific tool for decision-making in national disaster prevention and mitigation in disaster prevention and mitigation strategies research.

2 Gray fixed weight cluster analysis

2.1 Concept of gray fixed weight cluster (Liu et al. 2010)

Let $x_{ij}$ be observations of indicators $j$ about object $i$, and $f_{j}^{k} ( \bullet )$ is the whitening weight function for the $k$ subclass of the $j$ indicator, $i \in N = \{ 1, \ldots ,n\}$, $j \in M = \{ 1, \ldots ,m\}$, $k \in S = \{ 1, \ldots ,s\}$. If the weight $\eta_{j}^{k}$, $j \in M$, $k \in S$ of the $k$ subclass of $j$ indicator is irrelevant to $k$, that is, for any $k_{1} ,k_{2} \in S$, $\eta_{j}^{{k_{1} }} = \eta_{j}^{{k_{2} }}$, then we can remove $k$ of $\eta_{j}^{k}$ and denote $\eta_{j}^{k}$ as $\eta_{j}$, $j \in M$ and take

$$ \sigma_{i}^{k} = \sum\limits_{j = 1}^{m} {f_{j}^{k} } \left( {x_{ij} } \right)\eta_{j} $$

as the gray fixed weight cluster coefficient with the object $i$ belonging to the $k$ gray class. According to the value of the weighted clustering coefficient, we can classify the clustering objects.

2.2 The steps of the gray fixed weight cluster analysis

Gray fixed weight cluster analysis is carried out according to the following steps:

Step 1: Construct the typical whitening weight function for the $k$ subclass of $j$ indicator $f_{j}^{k} ( \bullet ),j \in M,k \in S$.

A typical whitening weight function (Liu et al. 2010 ) generally includes three types, as follows:

• 1. If the whitening weight function $f_{j}^{k} ( \bullet )$ does not include the third and fourth turning point $x_{j}^{k} \left( 3 \right),x_{j}^{k} \left( 4 \right)$, as shown in Fig. 1, then we claim $f_{j}^{k} ( \bullet )$ as the upper-limit measure of the whitening weight function, denoted as $f_{j}^{k} \left[ {x_{j}^{k} \left( 1 \right),x_{j}^{k} \left( 2 \right), - , - } \right]$.

  Fig. 1

  

  The upper-limit measure of whitening weight function

• 2. If the second and third turning point $x_{j}^{k} \left( 2 \right),x_{j}^{k} \left( 3 \right)$ of the whitening weight function $f_{j}^{k} ( \bullet )$ coincide, as shown in Fig. 2, then we claim $f_{j}^{k} ( \bullet )$ as the middle-limit measure of the whitening weight function, denoted as $f_{j}^{k} \left[ {x_{j}^{k} \left( 1 \right),x_{j}^{k} \left( 2 \right), - ,x_{j}^{k} \left( 4 \right)} \right]$.

  Fig. 2

  

  The middle-limit measure of whitening weight function

• 3. If the whitening weight function $f_{j}^{k} ( \bullet )$ does not include the first and second turning point $x_{j}^{k} \left( 1 \right),x_{j}^{k} \left( 2 \right),$ as shown in Fig. 3, then $f_{j}^{k} ( \bullet )$ is considered to be the lower-limit measure of the whitening weight function, denoted as$f_{j}^{k} \left[ { - , - ,x_{j}^{k} \left( 3 \right),x_{j}^{k} \left( 4 \right)} \right]$.

  Fig. 3

  

  The lower-limit measure of whitening weight function

Proposition 1 (Liu et al. 2010)

• 1. The upper-limit measure of the whitening weight function is as follows:

$$ f_{j}^{k} (x) = \left\{ {\begin{array}{*{20}c} {0,} \hfill & {x < x_{j}^{k} (1)} \hfill \\ {\frac{{x - x_{j}^{k} (1)}}{{x_{j}^{k} (2) - x_{j}^{k} (1)}},} \hfill & {x \in \left[ {x_{j}^{k} (1),x_{j}^{k} (2)} \right]} \hfill \\ {1,} \hfill & {x \ge x_{j}^{k} (2)} \hfill \\ \end{array} } \right. $$

And the image of this function is shown in Fig. 1.

• 2. The middle-limit measure of the whitening weight function is as follows:

$$ f_{j}^{k} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} \hfill & {x \notin \left[ {x_{j}^{k} \left( 1 \right),x_{j}^{k} \left( 4 \right)} \right]} \hfill \\ {\frac{{x - x_{j}^{k} \left( 1 \right)}}{{x_{j}^{k} \left( 2 \right) - x_{j}^{k} \left( 1 \right)}},} \hfill & {x \in \left[ {x_{j}^{k} \left( 1 \right),x_{j}^{k} \left( 2 \right)} \right]} \hfill \\ {\frac{{x_{j}^{k} \left( 4 \right) - x}}{{x_{j}^{k} \left( 4 \right) - x_{j}^{k} \left( 2 \right)}},} \hfill & {x \in \left[ {x_{j}^{k} \left( 2 \right),x_{j}^{k} \left( 4 \right)} \right]} \hfill \\ \end{array} } \right.. $$

And the image of this function is shown in Fig. 2.

• 3. The lower-limit measure of the whitening weight function is as follows:

$$ f_{j}^{k} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} \hfill & {x \notin \left[ {0,x_{j}^{k} \left( 4 \right)} \right]} \hfill \\ {1,} \hfill & {x \in \left[ {0,x_{j}^{k} \left( 3 \right)} \right]} \hfill \\ {\frac{{x_{j}^{k} \left( 4 \right) - x}}{{x_{j}^{k} \left( 4 \right) - x_{j}^{k} \left( 3 \right)}},} \hfill & {x \in \left[ {x_{j}^{k} \left( 3 \right),x_{j}^{k} \left( 4 \right)} \right]} \hfill \\ \end{array} } \right. $$

And the image of this function is shown in Fig. 3.

Step 2: Determine the clustering weight $\eta_{j}$, $j \in M$ of each indicator.

Step 3: Calculate the gray fixed weight cluster coefficient.

Using the whitening weight function $f_{j}^{k} ( \bullet ),j \in M,k \in S$, clustering weight $\eta_{j}$, $j \in M$, and $x_{ij} ,i \in N,j \in M$, as observations on indicators $j$ about object $i$, which is derived from Step 1 and Step 2, the gray fixed weight cluster coefficient $\sigma_{i}^{k} = \sum\nolimits_{j = 1}^{m} {f_{j}^{k} } \left( {x_{ij} } \right) \bullet \eta_{j} ,i \in N,k \in S$ is calculated.

Step 4: Classify all the objects.

If $\mathop {\hbox{max} }\limits_{1 \le k \le s} \left\{ {\sigma_{i}^{k} } \right\} = \sigma_{i}^{{k^{*} }}$, then object $i$ belongs to gray class $k^{ * }$.

3 Gray fixed weight cluster of rainstorm and flood disasters losses in China

3.1 Steps of the gray fixed weight cluster of rainstorm and flood disasters losses in China

In this paper, the data are from Yearbook of Meteorological Disasters in China (“Yearbook”) for the period of 2005–2010. We use the historical disaster data from 2004 to 2009 in “Yearbook” and seven affected indicators, which are affected area of crops (million ha), the area of no output (million ha), the affected population (million person-time), number of deaths including those missing (people), collapsed houses (million rooms), direct economic losses (billion RMB yuan), and we analyze the rainstorm and flood disaster losses in the Chinese mainland by the means of gray fixed clustering analysis. The steps of gray fixed weight clustering are as follows:

Step 1: Calculate the mean of each affected indicator

According to the statistical data of rainstorm flood affecting mainland China (excluding Hong Kong, Macao, and Taiwan) in 31 provinces, municipalities and autonomous regions from 2004 to 2009, and considering the regional area, population, economic development level, and other factors, we calculate the mean of each affected indicator in each administration. The results are shown in Table 1.

Table 1 The losses mean of each administrative region according to seven indicators from 2004 to 2009

Step 2: Determine the whitening cut-off value

When determining the whitening weight function, it is important and critical to determine the cut-off value. The approximate golden section method to determine the cut-off value is adopted in the following section.

First, we sort the means of all administrations of rainstorm and flood disasters according to the seven indicators. Taking the area of affected crops as an example, we select the mean of the area of affected crops of the 31 provinces from 2004 to 2009 as the middle-whitening cut-off value (63.9, as shown in Table 2). We increase the middle-whitening cut-off value by 60 % (that is, 63.9 × (1 + 60 %) = 102.2), and then we regard this value as a high-whitening cut-off value (102.2, as shown in Table 2). Finally, we decrease the middle-whitening cut-off value by 60 % (that is, 63.9 × (1 − 60 %) = 25.6). We then regard this value as a low-whitening cut-off value (25.6, as shown in Table 2). Other indicators are handled likewise. The results are shown in Tables 2, 3:

Table 2 Sorting the means and determining whitening cut-off value

Table 3 Sorting the means and determining whitening cut-off value

Step 3: Determine the typical whitening weight function

Step 3.1: Construct whitening weight functions $f_{j}^{k} ( \bullet )\left( {j = 1, \ldots ,7;\,k = 1,2,3} \right)$ for the 7 indictors and 3 gray classes:

$$ \begin{gathered} f_{1}^{1} \left[ {63.9,102.3, - , - } \right],f_{1}^{2} \left[ {25.6,63.9, - ,102.3} \right],f_{1}^{3} \left[ { - , - ,25.6,63.9} \right] \hfill \\ f_{2}^{1} \left[ {8.9,14.2, - , - } \right],f_{2}^{2} \left[ {3.6,8.9, - ,14.2} \right],f_{2}^{3} \left[ { - , - ,3.6,8.9} \right] \hfill \\ f_{3}^{1} \left[ {336.1,537.8, - , - } \right],f_{3}^{2} \left[ {134.5,336.1, - ,537.8} \right],f_{3}^{3} \left[ { - , - ,134.5,336.1} \right] \hfill \\ f_{4}^{1} \left[ {36.4,58.2, - , - } \right],f_{4}^{2} \left[ {14.6,36.4, - ,58.2} \right],f_{4}^{3} \left[ { - , - ,14.6,36.4} \right] \hfill \\ f_{5}^{1} \left[ {2.7,4.3, - , - } \right],f_{5}^{2} \left[ {1.1,2.7, - ,4.3} \right],f_{5}^{3} \left[ { - , - ,1.1,2.7} \right] \hfill \\ f_{6}^{1} \left[ {6.8,10.9, - , - } \right],f_{6}^{2} \left[ {2.7,6.8, - ,10.9} \right],f_{6}^{3} \left[ { - , - ,2.7,6.8} \right] \hfill \\ f_{7}^{1} \left[ {21.7,34.7, - , - } \right],f_{7}^{2} \left[ {8.7,21.7, - ,34.7} \right],f_{7}^{3} \left[ { - , - ,8.7,21.7} \right] \hfill \\ \end{gathered}$$

Step 3.2: Draw images of whitening weight function $f_{j}^{k} ( \bullet )\left( {j = 1, \ldots ,7;\,k = 1,2,3} \right)$, as shown in Figs. 1, 2 and 3.

Now, take $f_{1}^{1} \left[ { 6 3. 9, 1 0 2. 3, - , - } \right],f_{1}^{ 2} \left[ { 2 5. 6, 6 3. 9, - , 1 0 2. 3} \right],f_{1}^{ 3} \left[ { - , - , 2 5. 6, 6 3. 9} \right]$ as an example to explain the meaning of the image corresponding to each whitening weight function. The whitening weight function $f_{1}^{1} \left[ { 6 3. 9, 1 0 2. 3, - , - } \right]$ is shown in Fig. 1, and the location of $x_{j}^{k} \left( 1 \right)$ in Fig. 1 is 63.9, and the location of $x_{j}^{k} \left( 2 \right)$is 102.3. The whitening weight function $f_{1}^{ 2} \left[ { 2 5. 6, 6 3. 9, - , 1 0 2. 3} \right]$ is shown in Fig. 2, and the location of $x_{j}^{k} \left( 1 \right)$ is 25.6, the location of $x_{j}^{k} \left( 2 \right)$ is 63.9, and the location of $x_{j}^{k} \left( 4 \right)$ is 102.3. The whitening weight function $f_{1}^{ 3} \left[ { - , - , 2 5. 6, 6 3. 9} \right]$ is shown in Fig. 3, and the location of $x_{j}^{k} \left( 4 \right)$is 25.6 while the location of $x_{j}^{k} \left( 3 \right)$ is 63.9. Other figures of the whitening weight function are similar to the above.

Step 3.3: Establish an expression of the whitening weight function. The whitening weight functions are, respectively, as follows:

$$ f_{1}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < 63.9} \\ {\frac{{x - 63.9}}{{102.3 - 63.9}},} & {63.9 \le x < {\text{102}}{\text{.3}}} \\ {1,} & {x \ge 102.3} \\ \end{array} } \right.\quad f_{1}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {{\text{25}}{\text{.6}},{\text{102}}{\text{.3}}} \right]} \\ {\frac{{x - 25.6}}{{63.9 - 25.6}},} & {{\text{25}}{\text{.6}} \le x < 63.9} \\ {\frac{{102.3 - x}}{{102.3 - 63.9}},} & {63.9 \le x < 102.3} \\ \end{array} } \right. $$

$$ f_{1}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,63.9} \right]} \\ {{\text{1}},} & {0 \le x < 25.6} \\ {\frac{{63.9 - x}}{{63.9 - 25.6}},} & {25.6 \le x < 63.9} \\ \end{array} } \right.\quad f_{2}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < 8.9} \\ {\frac{{x - 8.9}}{{14.2 - 8.9}},} & {8.9 \le x < 14.2} \\ {1,} & {x \ge 14.2} \\ \end{array} } \right. $$

$$ f_{2}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {3.6,14.2} \right]} \\ {\frac{{x - 3.6}}{{8.9 - 3.6}},} & {3.6 \le x < {\text{8}}{\text{.9}}} \\ {\frac{{{\text{14}}{\text{.2}} - x}}{{{\text{14}}{\text{.2}} - {\text{8}}{\text{.9}}}},} & {{\text{8}}{\text{.9}} \le x < {\text{14}}{\text{.2}}} \\ \end{array} } \right.\quad f_{2}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {{\text{0}},{\text{8}}{\text{.9}}} \right]} \\ {{\text{1}},} & {{\text{0}} \le x < {\text{3}}{\text{.6}}} \\ {\frac{{{\text{8}}{\text{.9}} - x}}{{8.9 - 3.6}},} & {3.6 \le x < 8.9} \\ \end{array} } \right. $$

$$ f_{3}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < {\text{336}}{\text{.1}}} \\ {\frac{{x - {\text{336}}{\text{.1}}}}{{{\text{537}}{\text{.8}} - {\text{336}}{\text{.1}}}},} & {{\text{336}}{\text{.1}} \le x < {\text{537}}{\text{.8}}} \\ {1,} & {x \ge {\text{537}}{\text{.8}}} \\ \end{array} } \right.\quad f_{3}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {134.5,537.8} \right]} \\ {\frac{{x - 134.5}}{{336.1 - 134.5}},} & {134.5 \le x < 336.1} \\ {\frac{{537.8 - x}}{{537.8 - 336.1}},} & {336.1 \le x < 537.8} \\ \end{array} } \right. $$

$$ f_{3}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,336.1} \right]} \\ {{\text{1}},} & {0 \le x < 134.5} \\ {\frac{{336.1 - x}}{{336.1 - 134.5}},} & {134.5 \le x < 336.1} \\ \end{array} } \right.\quad f_{4}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < {\text{36}}{\text{.4}}} \\ {\frac{{x - {\text{36}}{\text{.4}}}}{{{\text{58}}{\text{.2}} - {\text{36}}{\text{.4}}}},} & {{\text{36}}{\text{.4}} \le x < {\text{58}}{\text{.2}}} \\ {1,} & {x \ge {\text{58}}{\text{.2}}} \\ \end{array} } \right. $$

$$ f_{4}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {14.6,58.2} \right]} \\ {\frac{{x - 14.6}}{{36.4 - 14.6}},} & {14.6 \le x < 36.4} \\ {\frac{{58.2 - x}}{{58.2 - 36.4}},} & {36.4 \le x < 58.2} \\ \end{array} } \right.\quad f_{4}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,36.4} \right]} \\ {{\text{1}},} & {0 \le x < 14.6} \\ {\frac{{36.4 - x}}{{36.4 - 14.6}},} & {14.6 \le x < 36.4} \\ \end{array} } \right. $$

$$ f_{5}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < {\text{2}}{\text{.7}}} \\ {\frac{{x - {\text{2}}{\text{.7}}}}{{{\text{4}}{\text{.3}} - {\text{2}}{\text{.7}}}},} & {{\text{2}}{\text{.7}} \le x < {\text{4}}{\text{.3}}} \\ {1,} & {x \ge {\text{4}}{\text{.3}}} \\ \end{array} } \right.\quad f_{5}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {1.1,4.3} \right]} \\ {\frac{{x - 1.1}}{{2.7 - 1.1}},} & {1.1 \le x < 2.7} \\ {\frac{{4.3 - x}}{{4.3 - 2.7}},} & {2.7 \le x < 4.3} \\ \end{array} } \right. $$

$$ f_{5}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,2.7} \right]} \\ {1,} & {0 \le x < 1.1} \\ {\frac{{2.7 - x}}{{2.7 - 1.1}},} & {1.1 \le x < 2.7} \\ \end{array} } \right.\quad f_{6}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < {\text{6}}{\text{.8}}} \\ {\frac{{x - {\text{6}}.8}}{{1{\text{0}}{\text{.9}} - {\text{6}}.8}},} & {{\text{6}}{\text{.8}} \le x < {\text{10}}{\text{.9}}} \\ {1,} & {x \ge {\text{10}}{\text{.9}}} \\ \end{array} } \right. $$

$$ f_{6}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {2.7,10.9} \right]} \\ {\frac{{x - 2.7}}{{6.8 - 2.7}},} & {2.7 \le x < 6.8} \\ {\frac{{10.9 - x}}{{10.9 - 6.8}},} & {6.8 \le x < 10.9} \\ \end{array} } \right.\quad f_{6}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,6.8} \right]} \\ {{\text{1}},} & {0 \le x < 2.7} \\ {\frac{{6.8 - x}}{{6.8 - 2.7}},} & {2.7 \le x < 6.8} \\ \end{array} } \right. $$

$$ f_{7}^{1} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x < {\text{21}}{\text{.7}}} \\ {\frac{{x - {\text{21}}{\text{.7}}}}{{{\text{34}}{\text{.7}} - {\text{21}}{\text{.7}}}},} & {{\text{21}}{\text{.7}} \le x < {\text{34}}{\text{.7}}} \\ {1,} & {x \ge {\text{34}}{\text{.7}}} \\ \end{array} } \right.\quad f_{7}^{2} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {x \notin \left[ {8.7,{\text{34}}{\text{.7}}} \right]} \\ {\frac{{x - 8.7}}{{21.7 - 8.7}},} & {8.7 \le x < 21.7} \\ {\frac{{34.7 - x}}{{34.7 - 21.7}},} & {21.7 \le x < 34.7} \\ \end{array} } \right. $$

$$ f_{7}^{3} \left( x \right) = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{x}} \notin \left[ {0,21.7} \right]} \\ {1,} & {0 \le x < 8.7} \\ {\frac{{21.7 - x}}{{21.7 - 8.7}},} & {8.7 \le x < 21.7} \\ \end{array} } \right. $$

Step 4: Gray fixed weight cluster analysis. The calculation process of the gray fixed weight cluster is as follows:

Step 4.1: Determine the clustering weight of each indicator $\eta_{j}$, $j \in M$. Since the cluster indexes have different meanings and dimensions, and furthermore, a great disparity in number, weights should be assigned to various cluster indexes in advance. The statistical loss data of the rainstorm and flood disasters include four aspects, namely, crop disaster situation (affected area, no output area), situation of population disaster (affected population, dead population), situation of house disaster (collapsed houses, damaged houses), and direct economic losses. Each aspect has equal weight. There are seven independent indicators, whose weights satisfy $\eta_{1} = \ldots = \eta_{6} = 0.125,\eta_{7} = 0.25$, where $\eta_{i}$ is the weight of the $i -$th indicator, $i = 1,2, \ldots ,7$.

Step 4.2: According to 1,302 historical disaster data (the missing data are regarded as zero) in “Yearbook” and the formulation $\sigma_{i}^{k} = \sum\nolimits_{j = 1}^{m} {f_{j}^{k} } \left( {x_{ij} } \right) \bullet \eta_{j}$ for calculating the gray fixed weight cluster coefficient, the disaster losses can be classified as high, medium, and low gray classes.

When $i = 1$,

$$ \begin{aligned} \sigma_{1}^{1} & = \sum\limits_{j = 1}^{7} {f_{j}^{1} } \left( {x_{1j} } \right) \bullet \eta_{j} = \left[ {f_{1}^{1} \left( {3.8} \right) + f_{2}^{1} \left( {0.6} \right) + f_{3}^{1} \left( {2.9} \right) + f_{4}^{1} \left( 0 \right) + f_{5}^{1} \left( 0 \right) + f_{6}^{1} \left( {0.1} \right)} \right] \\ &\quad \times 0.125 + f_{7}^{1} \left( {0.6} \right) \times 0.25 = 0 \\ \end{aligned} $$

Similarly, $\sigma_{1}^{2} = 0,\sigma_{1}^{3} { = }1$.So, $\sigma_{1} = \left[ {\sigma_{1}^{1} ,\sigma_{1}^{2} ,\sigma_{1}^{3} } \right]{ = }\left[ {0,0,1} \right]$, where $\sigma_{1}$ represents the gray fixed weight cluster coefficient in the Beijing area in 2004. The gray fixed weight cluster coefficient in each area is calculated in the same way. The results are shown in Tables 4, 5, 6, 7, 8, 9.

Table 4 Gray fixed weight cluster coefficient, maximum value and assessment of disaster in 2004

Table 5 Gray fixed weight cluster coefficient, maximum value, and assessment of disaster in 2005

Table 6 Gray fixed weight cluster coefficient, maximum value, and assessment of disasters in 2006

Table 7 Gray fixed weight cluster coefficient, maximum value, and assessment of disasters in 2007

Table 8 Gray fixed weight cluster coefficient, maximum value, and assessment of disaster in 2008

Table 9 Gray fixed weight cluster coefficient, maximum value, and assessment of disaster in 2009

Step 4.3: Determine the maximum of the gray fixed weight cluster coefficient.

The results of finding the maximum of the gray fixed weight cluster coefficient in each area from 2004 to 2009 are shown in Tables 4, 5, 6, 7, 8, 9.

Step 4.4: Classification.

According to the gray fixed weight cluster coefficient and its maximum value in Tables 4, 5, 6, 7, 8, 9, we assess the gray class for each administrative disaster loss, based on disaster losses of high, medium, and low gray classes. For example, by $\mathop {\hbox{max} }\limits_{1 \le k \le 3} \left\{ {\sigma_{1}^{k} } \right\} = \mathop {\hbox{max} }\limits_{1 \le k \le 3} \left\{ {0,0,1} \right\} = \sigma_{1}^{3} = 1$, Beijing in 2004 can be judged to belong to the low-class area of disaster losses. Similarly, the disaster assessment of each area from 2004 to 2009 can be obtained, as shown in Tables 4, 5, 6, 7, 8, 9.

3.2 The gray fixed weight cluster risk assessment of rainstorm and flood disaster losses in China

The assessment results of rainstorm and flood disaster losses in administrative districts of mainland China (except for Hong Kong, Macao and Taiwan) from 2004 to 2009 based on the gray fixed weight cluster could be obtained according to the gray fixed weight cluster coefficient, maximum value, and disaster assessment in Tables 4, 5, 6, 7, 8, 9. The results are shown in Tables 10, 11, 12, 13, 14, 15:

Table 10 Gray fixed weight cluster assessment of rainstorm and flood disasters in mainland in 2004

Table 11 Gray fixed weight cluster assessment of and flood disasters in mainland in 2005 mainland

Table 12 Gray fixed weight cluster assessment of rainstorm and flood disasters in mainland in 2006 mainland

Table 13 Gray fixed weight cluster assessment of rainstorm and flood disasters losses in mainland in 2007

Table 14 Gray fixed weight cluster assessment of rainstorm and flood disasters losses in mainland in 2008

Table 15 Gray fixed weight cluster assessment of rainstorm and flood disasters in mainland in 2009

Results of the gray fixed weight cluster analysis can be obtained from Table 10: There were 9 areas belonging to the high-class loss areas of rainstorm and flood disasters in 2004, in which Shandong was the most typical one. Henan and Hubei were widely affected; Henan and Hunan were badly affected in the output areas; Hunan and Sichuan had a large affected population; Sichuan and Yunnan had relatively large casualties; buildings were damaged much more seriously in Chongqing and Sichuan; Sichuan and Hunan had more direct economic losses. Jiangxi was defined as medium-class loss area of rainstorm and flood disasters, where the affected areas and population were relatively larger. Beijing and 20 other provinces were defined as the low-class loss areas of rainstorm and flood disasters.

Results of the gray fixed weight cluster analysis can be obtained from Table 11: There are 11 areas belonging to the high-class loss areas of rainstorm and flood disasters in 2005, in which Heilongjiang was the most typical one. Heilongjiang and Anhui had a very large affected area; Anhui and Hubei had a large zero-output area; Chongqing and Sichuan were much more seriously affected in the number of the population; Hunan and Sichuan had high casualties; collapsed and damaged buildings in Guangxi were more serious; Guangxi and Sichuan had much higher direct economic loss; Liaoning, Shandong and Chongqing belonged to the medium-class loss areas of rainstorm and flood disasters, Shandong and Chongqing had a larger affected population; the direct economic losses were greater in Liaoning. Beijing and 16 other areas belonged to the low-class loss areas of rainstorm and flood disasters.

Conclusion that can be drawn from Table 12: 7 areas were classified as high-class rainstorm and flood disaster losses, among which Jiangsu was representative. Heilongjiang and Jiangsu had more affected and non-output areas; Jiangsu and Hunan had a more affected population; the houses in Hunan were more seriously damaged; Sichuan, Guizhou, and Yunnan had more casualties; Jiangsu and Fujian had higher direct economic losses. There was no middle-class area of rainstorm and flood disaster losses. Twenty-four areas belonged to the low class of rainstorm and flood disasters losses, among which Beijing was considered typical.

Conclusions that can be drawn from Table 13: 12 areas belonged to the high class of rainstorm and flood disaster losses, among which Jiangxi is representative. The situation of disasters in 2007 was more severe, and there were 12 areas where rainstorms and floods caused very grave direct economic losses. The middle-lower Yangzi area, southwest China, Shanxi (Shaanxi), and Shanxi suffered major rainstorm and flood disaster implications. The affected areas in Anhui and Hubei were larger than other provinces; Anhui had more non-output areas; Anhui, Henan, Hubei, and Sichuan had more affected populations; Chongqing, Sichuan, Guizhou, and Yunnan had more casualties; collapsed houses were more serious in Anhui and Sichuan; damaged houses were more serious in Shanxi (Shaanxi) and Shanxi; Anhui and Sichuan had large, direct economic losses. Guangdong and Guangxi were the middle class of rainstorm and flood disaster losses, where there were more affected populations and casualties. Seventeen areas belonged to low class of rainstorm and flood disaster losses, among which Beijing was considered typical.

Conclusions that can be drawn from Table 14: 6 areas belonged to the middle class of rainstorm and flood disaster losses, among which Jiangxi was the typical one. Jiangxi and Hubei had more non-output areas; Hubei, Hunan, and Guangxi had more affected populations; Sichuan and Yunnan had more casualties; collapsed and damaged houses in Sichuan, Hunan, and Guangxi were more than those in other provinces; Hunan and Guangxi suffered large direct economic losses. Those areas that had no rainstorm and flood disaster losses belonged to the middle class. No rainstorm and flood disasters belonged to middle-class area. Twenty-five areas belonged to the low class, in which Beijing was the typical one.

Conclusions that can be drawn from Table 15: 8 areas belonged to the high class of rainstorm and flood disaster losses, in which Heilongjiang was the typical one. Heilongjiang and Hubei had larger affected and non-output areas; affected populations and casualties in Sichuan were more than other provinces; houses were more seriously damaged in Sichuan; Hunan and Sichuan had more direct economic losses. The areas that had no rainstorm and flood disaster losses belonged to the low class. Twenty-five areas belonged to the low class, in which Beijing was the typical one.

Overall, there are three main reasons for the high-level rainstorm and flood disaster losses: First, in some regions of China, the climate changes are abnormal, extreme weather events occur frequently, and meteorological disasters happen frequently and simultaneously, among which presents the characteristics of serious rainstorm and flood disasters. For example, the heavy rainstorm and flood disasters lasted in the Huaihe River Basin from June to July in 2007, leading to an affected crop area of 10,464,000 hectares, casualties of 1,467, and a direct economic loss of 84.47 billion RMB yuan, which could be defined as a year of serious rainstorm and flood disasters. Second, the remarkable monsoon climate in China leads to the concentrated precipitation in the summertime, the obvious inter-annual variability, and the uneven seasonal distribution. For instance, in the summer of 2008, heavy rainstorm and flood disasters happened in the Pear River Basin and the upper reaches of the Xiangjiang River; in autumn of same year, the most powerful autumn rain in South China since 1951 caused a serious flood disaster which led to an affected crop area of 6,682,000 hectares, casualties of 915, and direct economic loss of 65.18 billion RMB yuan. Third, the strength and intensity of rainstorm and flood disasters have a direct relationship with factors like the regional climate, topography conditions, geological characteristics, the number and density of the population, and financial situations. In addition, in June of 2005, Guangdong, Guangxi, and Fujian suffered from heavy rainstorm and flood disasters due to the warm moist air and weak cold air, as well as the strong precipitation in South China and mid-south of Jiangnan. Influenced by this, severe floods, landslides, mudslides, and other disasters happened in some regions, which caused 117 casualties, a missing population of 66 and a direct economic loss of over 18 billion RMB yuan.

In short, the gray clustering analysis from 2004 to 2009 shows that the rainstorm and flood disasters in China have features that include wide range, high frequency, strong hazards, and severe loss due to the climate and geographical conditions as well as social-economic factors. Apart from natural factors, there are some other reasons for the rainstorm and flood disasters: the monsoon climate, high-intensity precipitation, long durations, uneven spatial and temporal distribution, unreasonable use of natural resources, low standards in flood control and water conservancy, population, and economic growth. The feature of flood distribution in China is high in the east and low in the west; high in the coast and low in inland; high in lake and plain areas and low in plateau and hilly areas; high in the eastern and southern sides of mountains and low in western and northern.

3.3 Zoning map of the risk assessment of rainstorm and flood disasters in China

According to the results of the gray clustering assessment from 2004 to 2009 in China, we studied the dynamic zoning of rainstorm and flood disasters in various regions of mainland China, thus drawing a zoning map of the risk assessment of rainstorm and flood disasters. The high-class loss regions of rainstorm and flood disasters are shown in red, the middle-class loss regions in yellow, and the low-class loss regions in three colors: northwest China and Tibet in brown, eastern and southern China in light blue, and other areas in green. The zoning maps of the risk assessment of rainstorm and flood disasters in mainland China from 2004 to 2009 are, respectively, shown in Figs. 4, 5, 6, 7, 8, 9:

Fig. 4

Zoning map of disaster risk assessment in 2004

Fig. 5

Zoning map of disaster risk assessment in 2005

Fig. 6

Zoning map of disaster risk assessment in 2006

Fig. 7

Zoning map of disaster risk assessment in 2007

Fig. 8

Zoning map of disaster risk assessment in 2008

Fig. 9

Zoning map of disaster risk assessment in 2009

Conclusions from Fig. 4 to Fig. 9: Zoning maps of the risk assessment of the rainstorm and flood disaster losses dynamically show the severe impact of rainstorm and flood disasters in the Chinese mainland and also show the degree of influence of rainstorm and flood disasters losses suffered each year. For example, there are more red areas in 2005 and 2007, which intuitively represents that China suffered heavy rainstorm and flood disaster losses during those 2 years and relatively light rainstorm and flood disaster losses during the other 4 years, which are well consistent with historical disasters.

The research results show that rainstorm and flood disasters have aggravated hazards in most regions of the Chinese mainland; the results of rainstorm and flood disaster losses are consistent with actual rainstorm and flood disasters in each administrative district. Facts have proven that the relevant departments attached great importance to disaster prevention and mitigation work. They have started “comprehensive national disaster prevention and mitigation strategies research” based on the disaster assessment data provided by relevant departments and therefore supply references for making prevention and mitigation decisions. Many positive and effective measures have been taken to reduce the disaster losses to ensure sustainable development of the national and provincial economies, and good results have also been obtained. Consequently, it is necessary to discuss systematically the future trends of rainstorm and flood disaster risks in every administrative district.

3.4 Future trends of rainstorm and flood disaster risks in China

Based on Tables 4, 5, 6, 7, 8, 9, we can get the frequency of high, middle, and low class of disasters from the assessment of rainstorm and flood disasters from 2004 to 2009. The results are shown in Table 16.

Table 16 The frequency of high, middle, low class of rainstorm and flood disasters from 2004 to 2009

According to the frequency of high, middle, and low-class rainstorm and flood disasters from 2004 to 2009, we sort the frequency of the high-class rainstorm and flood disasters, as shown in Table 17. Then, we further analyze the high-class disaster areas, learning the most serious one, which is very helpful for focusing flood defending and fighting work.

Table 17 Ranking the frequency of high-class rainstorm and flood disasters in 2004–2009

Conclusions that can be drawn from Table 17: The frequency of rainstorm and flood disasters in Hunan is 6 times from 2004 to 2009, which is the most serious one in China. Hubei, Guangxi, Sichuan take the second place at 5 times; Jiangxi and Yunnan, 4 times; Shandong, Henan and Chongqing, 3 times; Heilongjiang, Jiangsu, Anhui, Fujian, Guangdong, Guizhou, and Shanxi (Shaanxi), 2 times; Shanxi, 1 time. This shows that rainstorm and flood disasters appeared in Hunan every year and that the situations are very grave. During 6 years, the disaster statistics are as follows: The crop-affected area is 9.458 million hectares; non-output area is 1,777 hectares; the affected population is 63.96.9 million, the casualties are 361, the number of collapsed houses is 503,000; the number of damaged houses is 146 million; the direct economic losses amount to 41.94 billion RMB yuan. Therefore, Hunan suffered the most serious disasters in China.

Through a comprehensive analysis of Tables 16 and 17, we can draw a conclusion that the rainstorm and flood disaster areas are relatively concentrated. Hereby, we can further determine the future flood risk trends as follows:

• 1. The rainstorm and flood disasters mainly concentrate in East China, Central China, South China and Southwest China; meanwhile, they appear less in the North, Northeast, and Northwest. East China, Central China, South China, and Southwest China are severely afflicted areas, followed by the Northeast and Northwest. North China is a light disaster area.

• 2. The provinces with a high frequency of 4–6 times of rainstorm and flood disasters are Jiangxi, Yunnan, Hubei, Guangxi, Sichuan, and Hunan, among which rainstorm and flood disasters happen most easily in Hunan, Hubei, Guangxi, and Sichuan. Therefore, they are always the most serious disaster areas, and the possibility of rainstorm and flood disasters occurring in the future is the greatest, and the consequent disaster situations the most serious, too.

• 3. Those provinces, such as Shandong, Henan, and Chongqing, which have 3 times as high frequency of rainstorm and food disasters, the probability of rainstorm and flood disasters occurring is relatively small. But with increasingly serious global warming and all kinds of extreme weather and climate events occurring, severe rainstorm and flood disasters are very likely to happen due to factors like climate, natural geographic conditions, precipitation distribution in time and space, frequency of occurrence, etc.

• 4. As for those provinces, such as Heilongjiang, Jiangsu, Anhui, Fujian, Guangdong, Guizhou, and Shanxi (Shaanxi), which have 2 times as high frequency, the years of rainstorm and flood disasters fluctuate, but the trend of occurrence still exists. During the flood season especially, such disasters will happen much more easily in coastal areas.

• 5. In north, northeast and northwest China, where most parts have a dry climate and less rain, it is less probable for rainstorm and flood disasters to occur, but under the influence of an abnormal climate, strong rainfall is still likely to happen. Moreover, some long and large-scale continuous rain or frequent heavy rain will also cause rainstorm and flood disasters, which should not be neglected. For example, rainstorm and flood disasters have even occurred in Shanxi, Heilongjiang, and Shaanxi in history.

4 Conclusions

This paper takes flood disaster data for 31 regions of mainland China of the past 6 years from 2004 to 2009, attempting to apply the improved gray fixed weight clustering evaluation results to the economic loss assessment of floods. The results show that rainstorm and flood disaster losses in each administration are in accordance with actual flood disaster losses and that according to the clustering results of the flood disaster losses, drawing a zoning map of the flood disaster assessment in mainland China dynamically shows the condition of flood disaster losses, and the map is intuitively clear and has a good effect. By providing a more intuitive reference for the economy and for prevention and mitigation of the region, there is potential value in the risk assessment of meteorological disasters.

The gray fixed weight cluster theory is an effective method to assess rainstorm and flood disaster losses, and the method could be commonly used in the assessment of meteorological disasters, but in order to improve the accuracy of loss evaluation results of rainstorm and flood disasters, it still needs to be explored further in some respects. For example, accurately determining affected index weights and rationally selecting cut-off values for whitening, etc., are for future study.