Urban flood-bearing vulnerability evaluation based on the moment estimate weighting and improved gray target model

Floods are one of the most widespread and costly natural disasters worldwide (Gu et al. 2020; Liu et al. 2022; Rentschler et al. 2022). According to the 2021 Global Natural Disaster Assessment Report, flood events had the highest frequency during 2000–2021, accounting for 51% of extreme weather events and 62.7% of the number of people affected. The impacts of rapid urbanization on urban hydrology are becoming more pronounced, increasing urban drainage vulnerability (Zhang et al. 2018; Shukla & Gedam 2019; Pang et al. 2022). Urbanization also leads to highly concentrated social assets and significantly increases flood vulnerability (Wing et al. 2020; Chen et al. 2021; Wang et al. 2023; Yang et al. 2023). Urban management is challenged with increasing urban flooding risks in flood prevention, hindering sustainable urban development (Kang et al. 2023; Li et al. 2023a; Zia et al. 2023). As an important tool for flood disaster prevention, urban flood vulnerability assessment has significant practical value in flood risk management to reduce urban flood damage.

On 20 July 2021, a world-shattering flooding event occurred in Zhengzhou City, China. This extreme flooding event resulted in severe inundation, greatly impacting the city (Zheng et al. 2022). The flood caused 380 deaths, affected 1.9 million people and induced direct economic losses of 40.6 billion yuan (MEMPRC 2022). The rainfall on the day (552.5 mm) accounted for 86.2% of the annual average rainfall (640.8 mm) in Zhengzhou City. The maximum hourly rainfall (201.9 mm) broke the historical record since the establishment of Zhengzhou Railway Station in 1951 and also set a new record for meteorological observations in mainland China (Zheng et al. 2022). This rainfall event triggered many types of flooding disasters, such as urban flooding, river overflow and dike breach, and emergency discharge from reservoirs, and exposed the inefficiency of urban flood risk management (Dong et al. 2022a; Tu et al. 2023; Yang et al. 2023). Therefore, it is important to construct an efficient and accurate flooding vulnerability assessment method in order to improve the disaster emergency response capacity of Zhengzhou City.

Urban flooding disaster is characterized by the extent of damage caused by heavy rainfall disaster-causing factors on the urban disaster-bearing body after urban flooding vulnerability exceeds a certain threshold. The disaster-bearing body shows a certain degree of resilience in response to flooding. Thus, flood-bearing vulnerability is a collection of urban socio-economic factors and disaster prevention facilities. Currently, there are various forms of vulnerability assessment methods. For example, Darlington et al. (2022) used historical flood data for the past 25 years in Calgary, Canada, combined with vulnerability indicators (flood loss, postsecondary education level, recent movers, etc.), methodology to predict future social vulnerability trends, and found that residential values in high flood hazard areas increase residential values faster than low flood hazard areas; Xing et al. (2023) used streetscape and remote sensing imagery to estimate urban building flood vulnerability in Hefei City, China, and found that multi-source datasets are fused to improve building vulnerability assessment; Zhang et al. (2022) used the coupled one- and two-dimensional hydrodynamic model MIKE FLOOD to simulate flood vulnerability of the city of Zhengzhou, China to simulate flood vulnerability of the city of Zhengzhou, China. The results showed that the scenarios with higher vulnerability were mostly short-term rainstorms with significant spatial and temporal distribution. In conclusion, the trend analysis method is not suitable for areas where effective historical data is lacking. Remote sensing image data are often limited by spatial and temporal accuracy. Storm water flood simulation models are complicated to construct and are mostly applicable to small spatial scales.

The indicator system method is widely used in flood vulnerability assessment because it examines the risk situation of heavy rainfall and flooding from multiple aspects, with easy access to data and strong operability. For example, Bucherie et al. (2022) developed a social vulnerability index assessment methodology for flooding in Ecuador using indicators for vulnerable groups, socioeconomics, health, and education, combined with principal component analysis and expert knowledge. The index driven by principal component analysis was found to indicate a higher level of relative vulnerability than the expert results. Langlois et al. (2023) constructed a flood vulnerability assessment framework incorporating indicators of current, internal, and biophysical vulnerability of cities, and its application in Indonesia showed that Banten, Jakarta, West Java, and Central Java are susceptible to flood-related damages. Yang et al. (2023) used population and GDP indicators to predict flood exposure risk in Chaohu City, China, under future scenarios, and found that population-intensive and GDP-intensive areas are more vulnerable to flood exposure than other areas.

The indicator system method utilizes hazard elements as evaluation factors, combines mathematical methods to determine weights, and finally, assigns a comprehensive evaluation value (Jibhakate et al. 2023). Indicator weight is one of the most important factors affecting vulnerability evaluation. Three methods are commonly used to determine weights. The subjective assignment method (Lyu et al. 2020; Ramkar & Yadav 2021; Ye et al. 2023) includes the analytical hierarchy process (AHP) and the order relationship analysis (G1) method. They have the advantage of expert experience but are greatly influenced by human factors. The objective assignment method (Xu et al. 2018; Ziarh et al. 2021; Ye et al. 2023) includes the entropy weight (EW) and the inter-criteria correlation (CRITIC) methods. They can avoid subjective arbitrariness, excluding subjective advantages. The comprehensive assignment method (Lai et al. 2015; Wang et al. 2018; Peng & Zhang 2022; Ye et al. 2023) includes the multiplier synthesis method and the empirical coefficient method. These assignment methods use the weighting method to integrate subjective and objective weights. It has been found that the multiplier effect of the multiplier synthesis method was larger with smaller indicator weights. The empirical coefficient method is subjective and can overcome this effect. Therefore, determining indicator weights is one of the problems to be solved in vulnerability assessment models.

The core idea of the gray target model (GTM) is to set an optimal goal under the condition that there is no standard model and to take the distance from each scheme to the bull's-eye as the evaluation value of the scheme (Jin et al. 2020; Qin et al. 2022; Zhu & Wen 2023). With its simple principle and clear evaluation process, the GTM has been widely used in disaster risk assessment, environmental evaluation, and multi-objective decision-making (Li et al. 2018; Jin et al. 2020; Tansar et al. 2023; van Schaik et al. 2023), with reasonable and accurate evaluation results and significant development prospects. Scale correlation inevitably exists between multiple indicators in the indicator system method (Jiang et al. 2019; Sun et al. 2022; Ji et al. 2023). Existing gray target models often adopt Euclidean distance to solve the bull's-eye distance and ignore this feature. Traditional models only use the optimal value as a single bull's-eye, leading to an insignificant order of the scheme advantages and disadvantages when the bull's-eye distance of multiple schemes is the same or similar. In actual evaluation, due to the complex relationship between the indicators, the single bull's-eye is insignificant in explaining the evaluation results.

In order to scientifically diagnose urban flood-bearing vulnerability and improve the defects of multi-indicator assessment methods, the urban flood-bearing vulnerability assessment system was proposed in this article consists of three parts: Based on the socio-economic factor of the city, a new urban flood-bearing vulnerability indicator system was constructed (Section 3.2); Based on the correlation coefficient matrix of the Mahalanobis distance (Du et al. 2017; Chen et al. 2023; Ji et al. 2023) and the relative bull's-eye distance, an improved gray target evaluation model was proposed for evaluating the vulnerability of different scenarios, which improved the insignificant order of advantages and disadvantages in the traditional method (Section 3.3); Using the idea of distance estimation, a distance estimation empowerment method integrating multiple weights was put forward, which overcomes the contingency and inexperience of the single weights (Section 3.4).

The city socio-economic data in this article was obtained from the Zhengzhou Municipal Bureau of Statistics (https://tjj.zhengzhou.gov.cn/). The road network data was obtained from the official website of OpenStreetMap (https://www.openstreetmap.org). The river network data was from the National Center for Basic Geographic Information (http://www.ngcc.cn/ngcc/).

As the urban economy develops, urban flooding vulnerability increases. Urban investment in disaster prevention and relief also increases. Thus, urban flood-bearing capacity is strengthened. Urban flood-bearing vulnerability is expressed as the resilience of the disaster-bearing body to reduce the impact of flooding (Zhang et al. 2018; Rentschler et al. 2022; Wang et al. 2023; Yang et al. 2023). At a higher urban economic level, urban flooding vulnerability is larger. At a stronger disaster prevention and relief capacity, urban flood-bearing vulnerability is lower. Therefore, these two elements interact and provide feedback, both characterized by the flood-bearing vulnerability. The urban flood-bearing vulnerability indicators (Table 1) were constructed by considering urban socio-economic development factors, such as urban socio-economics, population size, degree of development (Li et al. 2023a, 2023b; Yang et al. 2023), urban disaster prevention, and mitigation implementation level and subsurface (Wahalathantri Buddhi et al. 2016; Kim et al. 2021).

Traditional GTM have the following problems:

Traditional GTM use only the optimal value as a single target center. When target distances of multiple schemes are the same or close, the interpretation of evaluation results is insignificant. When the target distances from the optimal value are the same or close, the distances from the worst target center often differ significantly, resulting in significant differences in evaluating the superiority or inferiority of the schemes.

Indicator correlations inevitably exist in an evaluation system. Traditional GTM use the Euclidean distance to calculate target distances, ignoring indicator correlations. In contrast, the Mahalanobis distance considers the covariance between indicators, reflecting the deviation of two indicators from their respective means. The Mahalanobis distance also uses correlation coefficients to fully describe the degree of correlation between indicators.

If 1/n is considered as the weight of the indicators, the target distance is the square root of the sum of the weighted target center coefficients for each indicator. This assumes that the indicators are of equal importance, which may not accurately reflect the actual situation.

In the comprehensive evaluation system, correlations between indicators are inevitable, such as the urban economy and the level of integrated facility construction (Jiang et al. 2019; Sun et al. 2022; Ji et al. 2023). Traditional gray target models use Euclidean distance to solve the bull's-eye distance, ignoring the correlation between indicators. The Mahalanobis distance considers the covariance between indicators, characterizing the degree of deviation of two indicators from their respective means. The correlation coefficient is utilized to fully describe the degree of correlation between the indicators (Ji et al. 2023). Therefore, based on the correlation coefficient matrix, the bull's-eye distances from the program Sc_i to the positive and negative bull's-eyes were established, respectively.

It is necessary to add 1 to the denominator in Equation (7) to avoid a positive bull's-eye distance of 0.

Indicator weights are commonly the key to the vulnerability evaluation system (Bucherie et al. 2022). Existing comprehensive assignment methods are equalization methods combining the advantages of subjective or objective assignment and have the advantage of being widely applied (Jiang et al. 2019; Wen et al. 2021; Ye et al. 2023). However, they have the disadvantages of multiplier effect or subjectivity.

Moment estimation theory is a statistical method that uses sample moments to estimate overall moments (Adekunle et al. 2020; Dong et al. 2022b; He & Peng 2022). Moment estimation is not only applicable to scenarios with fewer samples, but is also not affected by large inter-sample deviations. Multiple weights make up a statistical sample, and the idea of moment estimation can be used to determine an overall mean as an integrated optimization weight. It reflects the relative importance in the indicator system and also integrates subjective experience and objective statistical advantages.

The subjective weight determination methods involved are AHP and G1. AHP is a multi-objective decision-making method that calculates the ranking of indicators through the fuzzy quantization method of qualitative indicators, and its determined weight is w_s1. The G1 method is a rational assignment of the degree of importance of indicators through the experts' evaluation of the ordinal relationship of each indicator at the same level, and its determined weight is w_s2.

The steps to solve the optimal weights are as follows:

From 2006 to 2008, the flood-bearing vulnerability index at Level II increased and then decreased. Due to the strengthened drainage pipe network in 2007, the flood-bearing vulnerability was significantly reduced. From 2009 to 2020, the vulnerability index was at Level III (the second highest) and showed a significant downward trend. In 2013, the vulnerability index increased, i.e., a short-lived phenomenon of reduced flood risk. This is mainly due to the decreased urban population density in 2013 relative to 2012 and the increased disaster preparedness indicators (such as the fiscal-to-budget ratio and the drainage pipe network density).

Among the 8 years, Guancheng District had the highest flood-bearing vulnerability, while Huiji District had the lowest vulnerability. Compared with Huiji District, Guancheng District was the main urban area with a high degree of urbanization. The general disaster prevention capacity reflected in the high greening rate and drainage pipe network construction contributed to its greatest flooding vulnerability. In particular, the pipe network construction in Guancheng District was the lowest in the five districts. In 2012, the ratio of local financial income to expenditure was also severely reduced, even falling below 100%. Therefore, Guancheng District is the top priority for flood prevention and mitigation in Zhengzhou City.

Rapid urbanization can cause destructive changes in the urban flood-bearing environment (Lu et al. 2023; Wang et al. 2023). From 2005 to 2020, the total scale of construction land in Zhengzhou City increased by 702 km²; the area of watershed wetlands decreased by about 30% (https://tjj.zhengzhou.gov.cn/). In 2020, the resident population increased from 7.4 million in 2008 to 12.8 million. Under the constant threat of sudden extreme rainstorms, urban flood-bearing bodies and disaster-causing mechanisms significantly changed. Strengthening the management of urban flooding vulnerability is the most convenient way to cope with the double test of extreme weather and rapid urbanization.

There are large differences in urban terrain, geographic location, elevation, and economic level. The development of urban flood resilience planning and design programs in line with their conditions can enhance the implementability and relevance of disaster prevention (Endendijk et al. 2023; Wang et al. 2023). This study proposed flood prevention planning recommendations for Zhengzhou City from three levels. (1) At the macro-city level, accurate urban flood risk maps should be prepared, and the connectivity of pipe networks and water systems should be strengthened. (2) At the meso-neighborhood level, it is important to improve the resilience of older settlements to storm flooding. (3) At the micro-household level, it is important to improve residents' own emergency response capacity and to actively implement disaster insurance.

Combining different weights and evaluation models, the effectiveness and advantages of the OCM-IGTM model proposed in this article were tested through single-factor comparative experiments. The combination schemes are shown in Table 3.

Their comparison results are as follows:

• (1) The urban flood-bearing vulnerability index by OCM-IGTM was close to the average of each evaluation method, with a small degree of dispersion. The correlations with the other seven evaluation methods were 0.982, 0.997, 0.959, 0.984, 0.974, and 0.994 (0.982 on average), respectively. It shows that the results of the OCM-IGTM model proposed in this article were reliable.

• (2) The urban flood-bearing vulnerability index by the OCM-mGTM and OCM-IGTM was generally consistent with the ‘maximum and minimum levels’ when indicator correlations were considered. However, OCM-mGTM did not account for the significance of programmatic differences due to dual target centers. Thus, urban flood-bearing vulnerability index variability was not captured.

• (3) The urban flood-bearing vulnerability index by the OCM-dGTM and OCM-IGTM was very similar, considering programmatic variability. However, OCM-dGTM did not incorporate indicator correlations, making the ‘maximum and minimum levels’ different from the results by most methods. Therefore, the credibility of the results obtained without considering indicator correlation was reduced.

• (4) The improved gray target was adopted as an evaluation model to test the differences in results with different weights. The weights affected the absolute values of resilience more than the relative values of resilience for different years. Therefore, the evaluation results with different weights revealed the relative true values of resilience. The single subjective weighting method was greatly influenced by human factors and had certain differences in the ranking of the schemes. The single subjective weighting method was greatly influenced by human factors and had certain differences in the ranking of the schemes, with OCM-IGTM results closer to the mean of all scheme, compared with WLA-IGTM.

In conclusion, the OCM-IGTM I model constructed in this article has the following three characteristics: (1) This model takes advantage of integrating multiple independent weighting methods to determine the optimal combination of weights. (2) The Mahalanobis distance is introduced to reflect the correlation between indicators. (3) This model also incorporates the concept of double-target criteria to represent the difference in evaluation results. These demonstrate the advantages of the OCM-IGTM model.

In order to accurately determine urban flooding vulnerability and address the defects in the index weights and GTM, this article proposed a vulnerability evaluation model for urban flooding based on the distance estimation and weighting method. The proposed model was applied to Zhengzhou City. The following main conclusions are drawn:

First, traditional gray target models have the problems of insignificant assessment of superiority and inferiority using a single bull's-eye and exclusion of indicator correlations. This article proposed the relative bull's-eye distance and introduced the Marginal distance. This can effectively address these problems and provide an important reference for improving the GTM.

Second, in order to improve the subjective and objective drawbacks of single assignment, this article proposed a method of determining optimal weights with minimizing weight deviation as the objective function based on the idea of moment estimation. This method can significantly improve the ‘multiplier effect’ and subjectivity in the existing combined weight scheme and provide insights into weight determination. However, the selection of the base weight set involved some subjective factors.

Finally, Zhengzhou City was taken as an example to demonstrate the proposed model. From the time scale of 2006–2020, the vulnerability to disasters in Zhengzhou City showed an upward trend. On the spatial scale (i.e., Zhengzhou's five districts), Guancheng District had the relatively highest urban flooding risk, and Huiji District had the lowest flooding risk. The imbalance between rapid urbanization and slow disaster prevention and mitigation construction in Zhengzhou was identified. Therefore, it is urgently needed to strengthen the resilient city construction in Zhengzhou.

This research was funded by the National Natural Science Foundation of China (52209038 and 52109038) and the National Key Research and Development Program of China (No. 2021YFC3200203). The authors thank the anonymous reviewers for their valuable comments.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.