Resilience is a physical concept, defined as the ratio of the energy absorbed by material to the volume before its rupture [1]. It indicates the capacity of material for absorbing energy in the process of plastic deformation and fracture [2]. Seismic resilience reflects the property that a material can absorb a large amount of energy to produce deformation without suffering damage under the action of vibration load [3]. The water supply system is the most important infrastructure in cities, and thus, its resilience evaluation has attracted extensive attention from scholars and engineers. However, as the seismic resilience is a newly emerging research topic derived from the theory of resilient cities, its meaning, research content, research methods and techniques have not been clearly defined. To fill this gap, this paper summarises and analyses the meaning and evaluation methods of resilience using more than 100 references on resilience collected from China and other countries. The aim is to clarify the scope and feasible schemes for research on seismic resilience of the water supply system.

Timmerma and Primm were the first to associate natural disasters with the concept of resilience. For the design of circulating water supply systems, Todini [4] proposed that the resilience of such water supply systems reflects their ability to resist excessive pressure and failure. By comparing simulated data with real-life data, Rose et al. [5] calculated the direct economic loss, indirect economic loss and the capital generated during the recovery period of lifeline engineering following natural disasters. Moreover, Rose developed the concept of disaster resilience of both the water supply system and the power supply system from the perspective of economics. Bruneau and Reinhorn [6] first proposed that resilience involves four attributes and four dimensions. These four attributes are robustness, rapidity, resourcefulness and redundancy; the four dimensions are physical dimension, service dimension, social dimension and economic dimension. These attributes and dimensions can be applied to any engineering system. According to the United Nations International Disaster Reduction Strategy [7], resilience is an important attribute of engineering systems when facing the threat of natural disasters. Syed and Jayant [8] proposed that improving architectural standards and awareness of disaster prevention can improve the resilience of engineering systems. Crowley and Elliott [9] introduced the concept of seismic resilience and the major resilience affecting factors by analysing the losses to engineering systems caused by earthquakes. Wang [10] and Fujita [11] discussed robustness and resilience, identifying robustness as the structural performance of a system which should meet norms and standards; they showed that resilience has many meanings, which mainly involve reducing the probability of accidents and improving resilience. Penny et al. [12] studied the seismic resilience of a city in Chile, and the results indicated that this design concept should be incorporated into urban planning and construction. Penny also pointed out that effective urban planning, including a reasonable layout of water supply and power supply systems, can improve seismic resilience.

After nearly 20 years of research on the concept of seismic resilience of water supply systems, the academic community holds that the resilience of water supply systems during earthquakes is reflected in three stages — resistance stage, absorption stage and recovery stage. The concept involves two aspects: (1) the resilience level of the seismic performance of the water supply system. In this aspect the resilience level is reflected in the resistance stage and the absorption stage. The resilience level at the resistance stage manifests as the probability of system failure (e.g., the vulnerability and earthquake damage rate of the water supply system). The resilience level at the absorption stage manifests as the degree of loss (e.g., function loss and economic loss) to the system caused by an earthquake. (2) The resilience level of the recovery capacity of the water supply system after an earthquake. The resilience level in this aspect is reflected at the recovery stage and manifests as the recovery time, recovery degree and recovery path after earthquakes.

Based on these aspects, this paper uses the following definition of the safety of the water supply system [13]: Under the action of an earthquake with the relevant fortification intensity level for the area, the earthquake damage level of the water supply system should not exceed the moderate damage level, and a 70% capacity of the water supply service function should be ensured after an earthquake. The recovery capacity of the water supply system after an earthquake is defined of the water supply system to restore the water service function to 70% or above of the normal level within 2–3 days using local resources and manpower. Although these two definitions cannot be directly applied to study the seismic resilience of water supply systems, the underlying concepts can be used to define the meaning of resilience. Comprehensive considering the two aspects of the concept of resilience shows that the seismic resilience includes seismic safety and post-earthquake recovery capacity. In other words, the seismic resilience of the water supply system of a city manifests not only in the ability of the system to resist and absorb earthquake disasters but also in the ability to quickly recover from earthquakes by only using limited resources. Resilience is a combination of these two abilities.

Albert et al. [14], Callaway et al. [15], Cohen et al. [16] and Paul et al. [17] abstracted the infrastructure network as a simplified model based on the graph theory. Moreover, they defined the key nodes in this network and established a resilience model by applying the water supply pressure on key nodes as criteria for measuring network resilience. Strogatz [18], Newman [19] and Boccaletti et al. [20] applied complex network theory to optimise the network topology of the water supply system. Jeon and O’Rourke [21], Wang et al. [22], Shi [23] and Tabucchi et al. [24] analysed the impact of the Northridge earthquake (with a magnitude of 6.6) on the water supply system of Los Angeles, USA, adopting the node negative pressure treatment method. They used GIRAFFE software to establish the corresponding seismic model for the water supply system and calculated the seismic reliability of important water consumption nodes in the water supply system. Based on the research results obtained by Toprak [25], Adachi [26], Mahmood [27] and Guo [28], Halfaya [29] evaluated the water supply capacity of the water supply system after an earthquake by analysing the size, characteristics and joint form of the pipeline. The research results showed that pipelines should be arranged using the frequency and intensity of earthquakes occurring in the region as criteria.

Regarding input criteria, scholars mostly used one or more indexes to evaluate the seismic resilience of the water supply system in a region. However, this cannot be applied to all water supply systems, which prevents comparisons. The seismic resilience evaluation is divided into the two aspects of seismic safety evaluation and post-earthquake recovery ability evaluation. Scholars tried to combine these two research directions to comprehensively evaluate the seismic resilience level of water supply systems. However, without proposing corresponding evaluation criteria, the resilience level of water supply systems cannot be compared (neither horizontally nor vertically). For a specific water supply system, how large seismic action is used as a reference input is key for evaluating and classifying the seismic resilience of a water supply system. Existing studies indicated that the corresponding pipelines should be equipped according to the frequency and intensity of earthquakes in the focal area. According to the

Wu et al. [32] and Mitrani et al. [33] developed evaluation software for evaluating the recovery ability of water supply systems. Davis [34] conducted extensive research on the seismic effects of the water supply system and systematically analysed the research findings on reliability and applicability of the water supply system obtained by Ballantyne et al. [35], Taylor [36], Shinozuka et al. [37], Markov et al. [38] and Hwang et al. [39]. Davis divided the service functions of the water supply system into five categories for comparing the changes of service functions before and after earthquakes. Li and Shen [40], Zhuang and Gao [41] and Zhou et al. [42] established a prediction model for the probability of earthquake damage to water supply systems based on Monte Carlo simulation and applied it to analyse the overall seismic performances of water supply systems. Liu et al. [43] applied modern combinatorial optimisation algorithms for optimising the seismic topology of water supply systems. Moreover, Liu compared and analysed the advantages and disadvantages of various algorithms through numerical examples, identifying the optimal algorithm a as criterion for evaluating the resilience level of water supply systems.

Corresponding evaluation criteria should also be established for each output item of the seismic resilience evaluation of water supply systems. At present, regarding relevant criteria, most studies conducted by scholars outside of China are based on comparisons before and after earthquakes in a specific same area or use water supply pressure as criterion to discuss how the resilience of water supply systems can be improved. However, in China, most studies follow the concept of anti-seismic design of buildings and evaluate the resilience level of their water supply system according to the criterion of “suffering no damage during small earthquakes, being repairable during moderate earthquakes and not collapsing during strong earthquakes”. This criterion can be assumed as a prerequisite or framework for evaluating resilience. However, the boundary among “suffering no damage”, “being repairable” and “not collapsing” should be clearly defined. Only then can classifications be carried out and different regions be compared. Currently, applied standard specifications limit the performance level of different engineering structures in service, which should closely match the performance level required by seismic resilience. Otherwise, an imbalance of meeting the resilience requirements without meeting the requirements in the current specifications will emerge; alternatively, the requirements in the current specifications may be met, but the resilience requirements may not be met. For example, in

As indicated by Todini, the most effective way for designing the pressure network of the water supply system is to apply loop topology. This can ensure sufficient water supply pressure in the system to overcome the impact of leakage points and effectively supply water for users. In contrast, in a tree-like distributed network, failure of the water supply system may exert a great impact on reliability because certain nodes will be out of service or only achieve poor service levels for a certain time. Therefore, adjusting the loop topology by adding pipes and closing valves is an effective way to improve the service level of the water supply system. Todini also proposed an index for measuring the ability of water supply systems to cope with failures. Prasad and Park [45] and Jayaram and Srinivasan [46] proposed improvements for specific shortcomings of the above mentioned index. Tierney and Trainor [47], Cagnan and Davidson [48], Scawthorn et al. [49], O’Rourke [50] and Cimellaro and Reinhorn [51] studied the interruption time, the restoration progress and how the interruption time of the water supply and power supply systems can be shortened during earthquakes. Based on the four attributes of resilience, Cutter et al. [52] established a set of evaluation indexes for water supply and power supply systems in communities and compared the resilience levels between different communities. Susan et al. [53], Jonas et al. [54] and Guo [55] analysed how the post-earthquake recovery time of the water supply system can be minimised from the three dimensions of maximising the use of groundwater sources, connecting to external reservoirs and rationing water supply. Based on the four dimensions of resilience, Wu et al. [56] proposed an evaluation index system for urban resilience. Pedcris and O’Mashahiko [57] proposed an evaluation index for resilience of urban communities based on the analytic hierarchy process.

At the start of the research on the resilience evaluation method of water supply systems, several scholars proposed to measure the resilience level of the water supply system by using the resistance of the water supply system to disturbance and the speed of system recovery to a balanced state as evaluation indexes. However, over the following 20 years, the research focus was separated into two aspects. On the one hand, the resilience level of the seismic safety of water supply system was evaluated based on the seismic performance of a water supply system. For this, vulnerability, earthquake damage rate, functional loss and economic loss were used as evaluation indicators; alternatively, the water supply network was used as research object, using the water pressure and reliability of key water nodes as evaluation indexes. On the other hand, the resilience level of the post-earthquake recovery ability of a water supply system was evaluated based on its recovery ability, using the recovery time, recovery speed, recovery path, recovery degree and input resources as evaluation indexes. It has been pointed out in the previous section that resilience evaluation indicates a comprehensive evaluation of the seismic safety and post-earthquake recovery ability of a water supply system. The next step is to determine the evaluation indexes for the seismic safety index and the post-earthquake recovery ability index. Evaluation indexes used to evaluate seismic safety include earthquake damage rate (_{1}), vulnerability (_{2}), functional loss (_{3}), economic loss (_{4}), water reserve of the reservoir (_{5}), water quality parameters (_{6}), reliability of key nodes (_{7}) and water pressure on key nodes (_{8}). Evaluation indexes for evaluating post-earthquake recovery ability include post-earthquake water supply capacity (_{9}), water supply interruption time (_{10}), recovery time (_{11}), recovery speed (_{12}), recovery path (_{13}), recovery degree (_{14}) and input resources (_{15}). These indexes involve two key issues. First, the above mentioned evaluation indexes are not independent of each other. For example, earthquake damage rate (_{1}) is related to functional loss (_{3}) and economic loss (_{4}), and recovery time (_{11}) is related to recovery speed (_{12}). How reasonable and operable evaluation indexes are selected, how each evaluation index is introduced into the evaluation model and how the influence weight of each index is considered are key issues to be studied in the future. Second, the introduction of key evaluation indexes is inadequate. For example, input resources (_{15}) (i.e., post-earthquake recovery ability) are closely related to the input of human and material resources, materials, equipment, organisational management and other factors. However, few studies currently focus on this index. Related evaluation models mostly define this index either by estimation or hypothesis. Regarding research on input resources (_{15}), the author once proposed a formula for the relationship between the total length of the water supply network and the number of workers on duty, as well as a formula for calculating the recovery time based on the post-earthquake recovery efficiency. How previous studies are applied on the resilience evaluation is also a key issue that needs to be addressed in the future.

Rossman [58] proposes a resilience index for the water supply system (_{α}_{β}_{γ}

These indexes can be used to evaluate the service level of a water supply system. Rossman conducted a case study on a small town in an earthquake-prone zone of Italy, using software EPANET to analyse its water supply system. The analysis involved the relationship between water supply demand and resilience index under the premise of a certain degree of water supply pressure and water quality requirements.

Miles and Chang [59] and Chang et al. [60] comprehensively addressed the four attributes of resilience, constructed an evaluation model for disaster losses and quantitatively evaluated the seismic resilience of the water supply system. They suggested that the relevance between lifeline projects is of great significance for resilience evaluation. Therefore, for improving the seismic resilience of lifeline projects, it is necessary to strengthen inter-departmental cooperation and information sharing.

Using spatial network topology, Barth quantitatively analysed infrastructure, e.g., roads, power transmission lines and communications facilities and presented the following model:

Regarding the seismic resilience evaluation of water supply systems, it is most important to select appropriate evaluation indexes (_{1}, _{2}, _{3}, ..., _{n}

Through a review of the literature, the meaning and evaluation methods of the seismic resilience of water supply systems have been summarised and analysed. The following four key issues have been identified:

The seismic resilience of the water supply system manifests not only as the ability of the system itself to resist and absorb earthquake disasters but also as the ability of the system to quickly recover after earthquakes using limited resources. Therefore, the seismic resilience of the water supply system involves the two aspects of seismic safety and post-earthquake resilience, and therefore, resilience should be a combination of these two abilities.

To arrive at more reasonable evaluation results of the seismic resilience of the water supply system and enable both horizontal and vertical comparisons, for the first time, it is proposed that seismic action with seismic fortification intensity level should be adopted as reference input. Based on this, a seismic resilience evaluation model of the water supply system is established, and grading classification is carried out. Corresponding evaluation criteria should be established for various indexes regarding seismic resilience evaluation of water supply systems. The existing standards and specifications limit the performance levels of different engineering structures currently in service, which should match the required behaviour levels of seismic resilience.

The evaluation indexes that can be used to define the seismic safety index and the post-earthquake resilience index are summarised. Inductive analyses of the evaluation indexes for seismic resilience of the water supply system showed that the evaluation indexes are mutually correlated to a certain extent. In addition, studies on important evaluation indexes are rare. How to select reasonable and operable evaluation indexes, introduce various evaluation indexes to the evaluation model and conduct in-depth research on important evaluation indexes are key topics that need to be explored in the future.

At present, the seismic resilience evaluation methods and models for water supply systems are limited by many problems. There is currently no method for comprehensively evaluating seismic safety and post-earthquake resilience. In this paper, resilience indexes comprise seismic safety index and post-earthquake resilience index.

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Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model