摘记Seismic risk assessment of complex transportation networks

[UMI] - 2014Stanford- Seismic risk assessment of complex transportation networks (M. Miller)

Chapter 1 Introduction

Transportation systems perform a critical role in the ability of a society to “bounce back” after a natural disaster, such as an earthquake. Resiliency can be defined as “spanning both pre-event measures that seek to prevent hazard-related damage and losses and post-event strategies designed to cope with and minimize disaster impacts”. 1 韧性定义
This dissertation extends a general event-based framework used by researchers for assessing seismic risk in a region 23456. 本文拓展了之前这4篇论文的区域地震风险评估框架，流程大致是 Initially, for each event, an earthquake scenario is sampled from distributions of magnitude and location. 首先去采样一个地震事件，来自于一定分布的烈度和地址 Next, realizations of ground-motion intensity values are computed at each component of interest, such as a highway bridge. 第二步，给出特定感兴趣组件，例如高速桥梁结构，的地面运动烈度值 The following step is sampling a damage state for each component based on the ground-motion shaking intensity. 接下来一步，是根据地震动烈度信息计算各个组件的损伤状态（的一个抽样） For studies considering networks, the network is then damaged at the locations where the components that carry or cross parts of it are damaged. 当需要考虑对网络系统的影响时，那么网络会由于这个组件所在地恰在或横跨地震所在地导致破化 To summarize the impact, a performance measure is calculated. 为了度量这种影响，一种性能指标计算来操作 In the end, the performance measure captures the regional losses or the functionality of the damaged network and helps the researcher understand the impact, a starting point for risk mitigation. 最后，这一度量指标代表了区域网络系统的损失或功能损伤，这可以帮助研究者搞明白地震影响，也就是一次减灾和风险评价的起点

This dissertation proposes computational methods for a reasonably comprehensive, probabilistic event-based risk assessment of large, complex transportation networks. It shows the applicability of these methods to understanding and mitigating the risk of a real-life example, the San Francisco Bay Area transportation network. Thus, this dissertation aids in estimating the e↵ects of many possible future earthquakes to transportation networks and the people and communities that rely on them, including quantifying the annual likelihood of various levels of loss.

Chapter 2 Event-based methods for seismic risk assessment of complex transportation networks w/ case study

We present methods for an event-based seismic risk assessment using reasonably comprehensive transportation network and network performance models. 事件地震风险评价，用合理复杂的交通网络和网络性能模型 We developed an efficient model with fixed demand and focusing on trips by car, and a high-fidelity model capturing variable demand, non-car travel modes, and interdependencies between the road and transit network for modeling the network performance after an earthquake. 开发了一种高效的“固定需求”模型（关注车辆行驶路途的），和一种高精度的“需求可变的、路网内联相关的”模型（无关于车辆行驶路途） We apply our framework to a case study of the San Francisco Bay Area, including extensive road and transit networks. Because the efficient model is so quick at estimating transportation-related losses after a future earthquake, we can assess the likelihood of losses from a wide range of possible events. On the other hand, the high-fidelity model offers a rare level of detail for understanding the impacts to travel behavior and accessibility, although it is computationally expensive. 两种模型的特质，到底用哪种合理呢？

2.1 Intro

After collecting and processing the case study data, we generate spatially-correlated ground-motion intensity maps. We then use these maps to generate realizations of damage to the bridges and other components in the transportation network. The component damage states enable generating the damage map for the full transportation network, including transit. Then, we represent the potentially damaged transportation network by two models: an efficient model that considers fixed demands and solely the road network; and a high-fidelity model that combines a variable demand model and various travel modes, such as walking, driving, biking, ride sharing, and taking mass transit. We use these models to compute various network-performance measure values. For each network-performance measure, we evaluate the annual likelihood of loss for different levels of performance. 这段又说明了2个模型的不同，可能我理解不到位，有空回来读

This work offers transportation planners the opportunity to transition from scenario by-scenario analysis to a broader range of possible events that could impact the transportation systems. The work also empowers researchers to model more of the complexities and interdependencies of infrastructure networks, while keeping the analysis computationally feasible.

2.2 Framework7

2.2.1 Ground motion Intensity maps

  • First, we generate QQQ earthquake scenarios from a seismic source model. The seismic source model specifies the rates at which earthquakes of specified magnitudes, locations, and faulting types will occur. This set of earthquake scenarios is comparable to a stochastic event catalogue in the insurance industry.
  • Second, for each earthquake scenario in the seismic source model, we use an empirical ground-motion prediction equation (GMPE) [e.g., 35–38] to model YYY , the resulting intensity measure at each location of interest [e.g., 39–41].
  • Then, for each of the QQQ earthquake scenarios, we sample bbb realizations of the spatially-correlated ground-motion intensity residual terms. Readers are referred to 3 for a survey of sampling methods. Once residuals are sampled, the total log ground-motion intensity (YYY) is computed as
    ln⁡Yij=ln⁡Y(Mj,Rij,Vs30,i,…)ˉ+σijϵij+τjηj\ln Y_{ij} = \bar{\ln Y\left(M_j, R_{ij}, V_{s30,i}, \dots \right)}+\sigma_{ij}\epsilon_{ij}+\tau_j\eta_jlnYij​=lnY(Mj​,Rij​,Vs30,i​,…)ˉ​+σij​ϵij​+τj​ηj​
    where jjj is the ground-motion intensity map index (j=1,…,mj = 1, \dots ,mj=1,…,m where m=Q×bm = Q \times bm=Q×b), ϵij\epsilon_{ij}ϵij​ is the normalized within-event residual in ln⁡Y\ln YlnY representing location-to-location variability and ηj\eta_jηj​ is the normalized between-event residual in ln⁡Y\ln YlnY and the other parameters are defined above. Both ϵij\epsilon_{ij}ϵij​ and ηj\eta_jηj​ are normal random variables with zero mean and unit standard deviation. The vector of ϵij\epsilon_{ij}ϵij​ can be modeled by a spatially-correlated multivariate normal distribution [e.g., 42] and the ηj\eta_jηj​ by a standard univariate normal distribution.
  • The result is a set of mmm ground-motion intensity maps (e.g., Figure 3.2(a)). Since we simulate an equal number (bbb) of ground-motion intensity maps per earthquake scenario, the annual rate of occurrence for the jjjth ground-motion intensity map is the original rate of occurrence of the earthquake scenario, divided by bbb. We denote the rate associated with the jjjth ground-motion intensity map as wjw_jwj​ .

Damage maps

The link between ground-motion intensity and structural damage is often provided by fragility functions. Fragility functions express P(DSi≥dsζ∣Yij=y)P(DS_i \ge ds _\zeta |Y_{ij} = y)P(DSi​≥dsζ​∣Yij​=y). We assume one component, such as a bridge, per site location, so we will identify both components and site locations via the index iii. Using that notation, DSiDS_iDSi​ is a discrete random variable whose value represents the damage state for the iiith component and dsζds_\zetadsζ​ is a damage state threshold of interest. The damage state is conditioned on a realization, yyy, of the random variable YijY_{ij}Yij​, the ground motion intensity at the iiith site and jjjth ground-motion intensity map. 易损性|条件地震烈度，对每桥（组件，component）

By sampling a damage state for each component, with probabilities obtained from the fragility functions given the ground-motion intensity, we produce a damage map. The damage map has a realization of the damage state of each relevant component. This sampling process can be repeated multiple times to simulate multiple damage maps per ground-motion intensity map. 对每桥（组件，conponent）损伤状态抽样后，得到损伤图

Functional percentage relationships link the component damage to the functionality of network elements. For example, in a road network, when a bridge collapses, the traffic flow capacity of the road it carries and it crosses can be modeled as reduced to zero. These relationships are typically derived from a combination of observation and expert opinion, often due to data scarcity 8. Furthermore, the relationships are typically deterministic for a certain component damage state and restoration time 8. Thus, in this paper, each damage map corresponds to a functionality state for every element of the network.

Network-performance measures and risk analysis

  • The final step for the event-based risk analysis is to evaluate the network performance measure, XXX.

We will consider various network performance measures, each of which is a scalar quantity per damage map. For road networks, two common performance measures are connectivity [23, 48] and flow capacity [49]. In order of increasing general computational cost, other measures to capture road network performance include the percentage of bridges damaged, weighted-shortest path between locations of interest [50], fixed-demand travel-time [18, 19, 51, 52], morning or evening peak commute time, economic impacts from increased travel time and bridge repairs [51], and mode destination accessibility [53]. Fixed-demand travel time delay and its variants have become particularly popular in current literature (e.g., Figure 3.2©). For power networks, connectivity is also a common network performance measure [26]. Recently, power network researchers have introduced other measures such as serviceability ratio [54], power system flow [55], and recovery time of the electrical network [56]. Researchers have also used connectivity to measure the reliability of water networks with alternative measures including flow capacity, entropy-based measures, nodal demands, and the total number of component failures system-wide [57–59]. 一段很好的综述

  • Once the chosen performance measure is computed for each damage map, understanding the exceedance rate of di↵erent levels of performance is a common goal.

The annual rate, λ\lambdaλ, of exceeding some threshold of network performance is estimated by summing the occurrence rates of all damage maps in which the performance measure exceeds the threshold:
λX≥x=∑j′=1Jwj′Π(Xj′>x)\lambda_{X \ge x} = \sum\limits^J_{j^\prime =1} w_{j^\prime} \Pi(X_{j^\prime} \gt x)λX≥x​=j′=1∑J​wj′​Π(Xj′​>x)
where xxx is a network performance measure threshold of interest and Xj′X_{j^\prime}Xj′​ is the network performance realization for the j′j^\primej′th damage map. The variable wj′w_{j^\prime}wj′​ is the occurrence rate of the j′j^\primej′th damage map as introduced above. The indicator function evaluates to 1 if the argument, Xj′>xX_{j^\prime} \gt xXj′​>x, is true, and 0 otherwise. By evaluating λ\lambdaλ at different threshold values, we derive an exceedance curve.

Procedure for aggregating interdependent infrastructure risk data

Key steps are as follows:

  1. Study area Identify study area and determine networks to model.
  2. Component locations
    For each network in the study area, select components where seismic damage
    will be modeled. For example, bridges and other above-ground structures may be particularly vulnerable parts of a transportation network.
  3. Hazard
    For all selected components, gather relevant hazard data. For seismic hazard, this entails:
    • Site conditions: At each component location, characterize the site conditions, such as summarized by Vs30,i values. Readily-available approximate methods may be more tractable than finding these values from field studies [e.g., 60].
    • Source model: Choose a seismic source model to describe possible earthquake scenarios via magnitude, rupture location, rupture type, and fault, and their respective annual rates of occurrence.
    • Ground-motion model: Simulate the level of shaking at each location identified in Step 2 for each earthquake event. Example methods include ground-motion prediction equations, or a physics-based hybrid-broadband model. A model for spatial correlation of the intra-event residuals is also needed if ground-motion prediction equations are used at this step.
  4. Component data
    For all selected components, determine relevant interaction data.
    • Fragility data: Fragility functions estimate each component’s damage state given a level of ground-motion shaking intensity. Examples include independent functions each in the form of a lognormal distribution [61], time-dependent models [62], and multivariate models that capture the correlation between components [e.g., 45, 46].
    • Functionality data: When a component is damaged, functional percentage relationships specify the functionality of the associated network element, e.g., road segment.
  5. Network topology
    For each network, collect data about topology (as a graph). Network data, i.e. graph data, can be formulated in various ways such as a list of edges, an adjacency matrix, or a list of nodes with edge information as an attribute of the node. It is important that a consistent format is chosen for all networks, which may require conversion. Note that this network can also be estimated with electronic data, such as by using geo-tagged mobile phone and GPS data to map roads [e.g., 63].
  6. Network interdependency
    Determine the locations where the networks are interdependent and formalize this interaction. For example, people may switch from a bus to a train at a train station, which translates to a specific node in the bus network that is physically adjacent to a node in the train network. Or, for example, the electric network may interact with the water network as power for a water pump. If the interaction points are not readily available, aerial imagery may be useful.
  7. Network damage from component damage
    Match selected network components to the relevant network graph. A general procedure is to create a hash table or look-up table between relevant components (identified in Step 2) and corresponding edges of each network (determined in Step 5). For example, a highway bridge may carry a highway road and may cross a train track and a local road. Aerial imagery may also aid this matching.
  8. Network performance measure
    Determine how network performance will be measured and build or find a relevant model or models for measuring network performance.
  9. Performance-measure model
    Gather relevant additional data for the network-performance models, such as:
    • Network edge properties: Many models require properties of the edges of the graph(s), such as flow capacity or length. If this data is not readily available, it can be estimated through aerial imagery, design drawings, and expert opinion. Furthermore, mobile data has been shown to provide good estimates for some networks, such as road networks [64].
    • Impact models: Impact can be measured in dollars, deaths, or downtime [65]. As such, depending on the network performance model, information is needed about the direct monetary costs (i.e., repair costs) and indirect monetary costs (e.g., value of travel time or cost of business interruptions), impact on human lives, or downtime (e.g., data about the restoration time of damaged components).
    • Links from network state to impact measures: The interaction between the measures of impact and the graph of nodes and edges must be formalized. For example, if there is a certain flow of people between two cities, to which nodes in the graph G, do the origin and destination cities respectively correspond?

2.4 Case Study

2.4.1 Network description and model representation

2.4.2 Ground motion intensity map models

2.4.3 Damage maps

Transit network damage

Each of the 43 transit systems we considered will be impacted di↵erently. For Caltrain, conversations with managers suggest that given that there is one shared track system, the system would either be fully operational or not at all. Similarly, managers suggested modeling the VTA system as fully functional or not. Depending on where the BART train cars are when the earthquake strikes, the agency could accommodate di↵erent emergency plans. However, BART representatives suggested considering that if any part of a route is damaged, the entire corresponding route would not be operational (but other routes on di↵erent tracks might be still operational). In other words, each BART route as well as the Caltrain and VTA routes are each a weakest-link system, so the failure of a single component will cause the route to be non-operational. We modeled the ferry systems as fully functioning for all earthquake events. For all earthquake events including the baseline, trans-bay and cross-county bus lines were discontinued, but main lines in urban areas as well as other local bus networks were maintained per recommendations from the MTC, though they may face delays due to traffic congestion.

Road network damage

Each component damage state maps directly to the traffic capacity on associated road segments. We use a functional percentage relationship to compute the traffic capacity of relevant road segments. Based on discussions with Caltrans, we consider travel conditions one week after an earthquake, since it is a critical period for decision making. For example, one week after most events, bridges should have been inspected and surface damage should be repaired, but major reconstruction would not have yet begun. According to our functional percentage relationship, at this point in time, the components have one of two classes of functionality, zero traffic capacity and full traffic capacity 8. We can thus summarize the component damage using two damage states dss, dsdamaged and dsfunctional, which correspond to the common HAZUS extensive or complete damage states and the none, slight, or moderate damage states respectively 8. Thus, the functional percentage relationship assigns zero traffic capacity on road segments that have at least one component in the dsdamaged damage state, and full traffic capacity otherwise. We do not consider network damage from sources other than main structural damage from ground shaking, such as tunnel displacement or liquefaction, but the framework allows including such considerations. In the discussion below, we consider a set of 113,940 damage maps, which correspond to 2110 scenarios, 3 ground-motion intensity maps per scenario, and 18 damage maps per ground-motion intensity map.

2.4.4 Network-performance measures

For each damage map, we aim to measure the network performance. Networkperformance measures can be classified into four levels: (Level I) vulnerability analysis, (Level II) connectivity analysis, (Level III) flow analysis and (Level IV) serviceability analysis 9.

  • Vulnerability analysis (Level I) refers to estimating the earthquake-related losses to the set of components. For the case study, we implement two network performance measures for vulnerability analysis: percentage of highway components damaged=100∑i∈BhΠ(dsi,j′≥dsext)∣Bh∣=100\frac{\sum_{i \in B_h}\Pi \left(ds_{i,j^\prime}\ge ds_{ext}\right)}{|B_h|}=100∣Bh​∣∑i∈Bh​​Π(dsi,j′​≥dsext​)​ and percentage of all components damaged=100∑i=1nΠ(dsi,j′≥dsext)n=100\frac{\sum^n_{i =1}\Pi \left(ds_{i,j^\prime}\ge ds_{ext}\right)}{n}=100n∑i=1n​Π(dsi,j′​≥dsext​)​, which we will hereafter simply call percentage of all components damaged. Where BhB_hBh​ is the set of indices of the highway components, ∣Bh∣|B_h|∣Bh​∣ is the size of this set (1743), nnn is the number of all components (3152), and dsi,j′ds_{i,j^\prime}dsi,j′​ represents the realization of the damage state for the iiith component in the j′j^\primej′th damage map using fragility functions and the inverse method as described in Section 2.2. As defined in Section 2.2, dsextds_{ext}dsext​ is the extensive damage state, and the function is an indicator function that evaluates to 1 if the argument, dsi,j′≥dsextds_{i,j^\prime} \ge ds_{ext}dsi,j′​≥dsext​, is true and 0 otherwise.
  • Connectivity analysis (Level II) captures which nodes will be disconnected. However, since the graph representing the case study network is very densely connected with many redundant paths and we model many local roads as not damaged (Section 2.4.3), it is highly unlikely that two nodes will be disconnected. Thus, we do not implement any connectivity analysis metrics for the case study.
  • Flow analysis (Level III) captures the flow between different nodes [e.g., 49] to capture the complexities missed by vulnerability analysis. Two cases of a maximum flow network-performance measure: from the financial district in San Francisco to a residential area in San Francisco (Figure 2.13(a)), and from the San Francisco area to the Oakland area (Figure 2.13(b)). Specifically, given the graph G=(V,E)G= (V, E)G=(V,E), let s,t2 V be the source and sink respectively, and let the capacities of each edge, cf (u, v) between nodes u and v, represent the maximum amount of traffic that can pass along a road segment (edge) per unit time. Then, the flow, f, is subject to two constraints:

In other words, the flow on each edge must be less than the capacity and there is conservation of flow at each node (flow in = flow out).
The maximum flow occurs when the total flow from the source is maximized. In other words, we maximize F=∑v:(s,v)∈Ef(s,v)F =\sum\limits_{v:(s,v)\in E} f(s,v)F=v:(s,v)∈E∑​f(s,v) For the case study, we have computed this maximum flow value using the Ford-Fulkerson algorithm [93] implemented in the Python package NetworkX [72], as part of the graph corresponding to our efficient network model. To obtain more robust results, we introduce supernodes at each source and sink [75] (Section 2.4.1).

  • Serviceability analysis (Level IV) performance measures: travel time increase, vehicle-miles traveled increase (VMT), and mode-destination accessibility.

2.4.5 Mode-destination accessibility

Mode-destination accessibility, hereafter referred to as accessibility, measures the distribution of travel destination opportunities weighted by the composite utility of all modes of travel to those destinations, i.e., the ease of someone getting to different destinations weighted by how desirable those destinations are [53, 82, 101]. The utility function for the mode-destination choice may be estimated using a multinomial random utility model where the logsum represents the accessibility value.

Chapter 3 Ground-motion intensity and damage map selection for probabilistic infrastructure network risk assessment using optimization10

Here we show a computationally-efficient method for selecting a subset of damage maps, corresponding ground-motion intensity maps, and associated occurrence rates using optimization that reasonably estimates the full distribution of a target performance measure and the ground-motion intensity. From a set of candidate damage and ground-motion intensity maps and corresponding realizations of network performance, the optimization problem involves choosing a reduced set of damage and ground-motion intensity maps and associated annual rates of occurrence that minimizes the differences between the marginal ground-motion intensity and network performance loss exceedance curves of this set and an extensively-sampled baseline set over a range of return periods of interest. The optimization procedure implicitly includes the joint distribution of the ground-motion intensity via the network performance measure.

Chapter 5 Prioritizing bridge retrofits using network importance and seismic risk

In this chapter, we present a method for combining network importance and seismic risk into a composite importance measure for prioritizing bridge retrofit evaluation during preliminary screening. We calculate this importance measure using the annual rate of bridge damage conditioned on a high loss in transportation network performance. We demonstrate that rankings based on this importance measure are not particularly sensitive to the high loss threshold, nor to the network-performance measure, although the method requires a reasonably extensive set of possible earthquake events. Finally, we show that the proposed method captures joint effects between expected damage and network performance, which may not be captured by these two factors independently.

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  2. Jayaram N, Baker JW. Efficient sampling and data reduction techniques for probabilistic seismic lifeline risk assessment. Earthquake Engineering & Structural Dynamics 2010; 39(10):1109–1131, doi:10.1002/eqe.988. ↩︎

  3. Han Y, Davidson RA. Probabilistic seismic hazard analysis for spatially distributed infrastructure. Earthquake Engineering & Structural Dynamics 2012; 41(15):2141–2158, doi:10.1002/eqe.2179. ↩︎ ↩︎

  4. Grossi P, Kunreuther H. Catastrophe Modeling: A New Approach to Managing Risk. Springer, 2005. ↩︎

  5. Johnson L, Samant LD, Frew S. Planning for the Unexpected: Land-Use Development And Risk. American Planning Association, 2005. ↩︎

  6. Bommer J, Spence R, Erdik M, Tabuchi S, Aydinoglu N, Booth E, del Re D, Peterken O. Development of an earthquake loss model for Turkish catastrophe insurance. Journal of Seismology Jul 2002; 6(3):431–446, doi:10.1023/A:1020095711419. ↩︎

  7. M. Miller and J. W. Baker, “Ground-motion intensity and damage map selection for probabilistic infrastructure network risk assessment using optimization,” Earthquake Engineering and Structural Dynamics, 2014. ↩︎

  8. Werner S, Taylor C, Cho S, Lavoie J. REDARS 2 methodology and software for seismic risk analysis of highway systems (technical manual). Technical Report, Seismic Systems & Engineering Analysis for MCEER, Oakland, CA 2006. ↩︎ ↩︎ ↩︎ ↩︎

  9. Kakderi K, Alexoudi M, Argyroudis S, Kyriazis P. Definition of system components and the formulation of system to evaluate the performance of water and waste-water systems. SYNER-G Deliverable 2.5, Joint Research Centre of the European Commission, European Union 2011. ↩︎

  10. M. Miller and J. W. Baker, “Ground-motion intensity and damage map selection for probabilistic infrastructure network risk assessment using optimization,” Earthquake Engineering and Structural Dynamics, 2014. ↩︎

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