Prior to the development of a methodology or the expense of any time or resources, it is considered appropriate to review the existing literature pertaining to shockwaves. There are three key aims of this literature review: 在制定方法或花费任何时间或资源之前，我们认为应该审查与冲击波有关的现有文献。本次文献审查有三个主要目的：
To capture the findings of other studies relating to shockwaves, and to a limited extent evaluate their validity. 收集与冲击波有关的其他研究结果，并在一定程度上评估其有效性。
To allow those findings that are found to be valid and relevant to shape the path of this project. 让那些被认为有效和相关的研究结果决定项目的发展方向。
To demonstrate knowledge and understanding of previous studies. 展示对以往研究的了解和理解。
This review will begin with a description of the concepts of density, flow and speed as they are commonly understood. It will be seen that such concepts are key to the remainder of this paper, hence their inclusion at its outset. 本文将首先介绍人们通常理解的密度、流量和速度概念。我们将看到，这些概念是本文其余部分的关键，因此在本文一开始就将它们包括在内。
The seminal macroscopic shockwave theories of Lighthill and Whitham, and Richards will then be described. They are widely considered to have initiated the comprehensive field of traffic theory, but being imperfect, the subsequent second order and second order resurrection theories are outlined to detail the legacy of the macroscopic approach. 然后将介绍莱特希尔和惠瑟姆以及理查兹的开创性宏观冲击波理论。这些理论被广泛认为是交通理论综合领域的开创者，但并不完善，因此我们将概述后续的二阶和二阶复活理论，以详细介绍宏观方法的遗产。
The comparatively broader church of car following models is covered with reference to four competing approaches. A quantitative comparison of these models is related and tentative conclusions are drawn. The ability of such models to generate shockwaves is then checked with reference to a theoretical paper by Richards. 通过参考四种相互竞争的方法，介绍了相对更广泛的汽车驾驶模式。对这些模型进行了定量比较，并得出了初步结论。然后，参考理查兹的一篇理论论文，检验了这些模型产生冲击波的能力。
Lastly, the relationship between shockwaves and accidents is described, drawing on a relatively new batch of studies. 最后，根据一批相对较新的研究，介绍了冲击波与事故之间的关系。
2.1 The Flow Density Curve 2.1 流量密度曲线
Density is understood to be the number of vehicles occupying a given length of roadway at a particular instant and is calculated as: 密度被理解为在某一特定瞬间占用一定长度道路的车辆数量，计算公式为
Where Density, Speed and Flow, the number of vehicles passing a point during a certain time interval. 其中， 密度、 速度和 流量，即在一定时间间隔内通过某点的车辆数。
Figure 2.0 shows the relationship of flow and density on a highway. There are two points at which flow is zero; this is at the origin, where density is zero - the highway is not trafficked (speed is theoretical at this point and would be determined by the first driver at a rate to the left of the curve, marked for example as Sf here); and at point Dj, where vehicles are so densely distributed along the highway that they are stationary. The density at which all vehicles are stationary is known as 'jam density'. 图 2.0 显示了高速公路上流量与密度的关系。有两个点的流量为零；一个是在原点，密度为零，即公路上没有车辆通行（此时的速度为理论速度，由第一位驾驶员以曲线左侧的速度决定，例如此处标记为 Sf）；另一个是在 Dj 点，车辆在公路上密集分布，处于静止状态。所有车辆都静止不动的密度称为 "堵塞密度"。
D0 is the density at which flow is highest, and this can be understood as the capacity of the highway - that is the maximum sustainable flow at which vehicles can pass a certain point under prevailing conditions. The slope of the line SO shows the speed of vehicles at full capacity, which may be called the critical speed. The curve to the right of DO is broken to denote over-capacity conditions, where additional vehicles increase density as opposed to flow. D0 是流量最大时的密度，可以理解为公路的通行能力，即在当时的条件下，车辆可以通过某一点的最大持续流量。SO 线的斜率表示车辆在满载情况下的速度，也可称为临界速度。DO 右侧的曲线断开，表示超过通行能力的情况，此时新增车辆增加的是密度，而不是流量。
The relationship between flow and density as demonstrated is likely known to readers intuitively or else from experience; as more vehicles traffic the highway headways reduce and it is more difficult to drive at higher speeds safely. Once a maximum sustainable speed is reached, additional vehicles have the effect of reducing speeds and therefore flow. The relationship as demonstrated has been empirically observed. A study of the A43 in Germany restated in Nagel, Wagner and Woesler (2003) concretises the fundamentals of the speed density curve. The results are charted in Figure 2.1, where is flow and is density. 读者很可能凭直觉或经验了解到流量与密度之间的关系；随着车辆的增加，高速公路的车头间距缩短，更难以安全地以更高的速度行驶。一旦达到可持续的最高车速，新增车辆就会降低车速，从而减少流量。这种关系已被经验所证明。Nagel、Wagner 和 Woesler（2003 年）对德国 A43 高速公路进行的一项研究证实了速度密度曲线的基本原理。结果如图 2.1 所示，其中 为流量， 为密度。
Flow and density appear to have a simple relationship up to the region where we expect to find maximum capacity. From this point on the data points are more widely dispersed. This dispersal is a result of a more complex relationship between and , resulting from increased interactions between vehicles. Such a wide scatter of data points in the congested region suggests a function with additional parameters and a dependence on understanding how vehicles interact. The various car following models which attempt to explain this condition will be discussed in Section 2.4 in order to give a full picture of the effects of density with respect to flow. 流量和密度之间似乎存在着一种简单的关系，直至我们预期发现最大容量的区域。从这一点开始，数据点更加分散。这种分散是 和 之间更复杂关系的结果，是车辆之间相互作用增加的结果。拥堵区域的数据点如此分散，表明该函数具有更多参数，并依赖于对车辆如何相互作用的理解。第 2.4 节将讨论试图解释这种情况的各种跟车模型，以便全面了解密度对流量的影响。
Figure 2.1 - Flow, Density and Congestion Source: Nagel, Wagner and Woesler (2003) 图 2.1 - 流量、密度和拥堵情况 资料来源：Nagel、Wagner 和 Woesler（2003 年）：Nagel、Wagner 和 Woesler (2003)
2.3 Macroscopic Shockwave Theories 2.3 宏观冲击波理论
The density flow curve for a particular highway link will vary over time, according to the proportion of commercial vehicles, the day of week and driver familiarity with the route. By assuming a single applicable density flow curve for a particular link at a particular time we can discuss the range of speeds, densities and flows we would expect to find. 某条高速公路的密度流量曲线会随着时间的推移而变化，这取决于商用车辆的比例、每周的哪一天以及驾驶员对路线的熟悉程度。通过假设某一特定时间某一特定路段的单一适用密度流量曲线，我们可以讨论我们预计会发现的车速、密度和流量范围。
2.3.1 Lighthill and Whitham 2.3.1 莱特希尔和惠瑟姆
Lighthill and Whitham (1955) find that the distribution of density along a highway link is largely continuous, since vehicles are driven by individuals and real speed differentials are present. However we can understand densities in terms of 'bunches' - multiple aggregations of densities. When a high density bunch meets with a low density bunch, which is made possible by inverse relationship between speed and density, then a "bunch of continuous waves can coalesce into a discontinuous wave, or 'shock wave'. When vehicles enter this condition their mean speed is substantially reduced very quickly." (Lighthill and Whitham 1955 pg. 43) Lighthill 和 Whitham（1955 年）发现，高速公路沿线的密度分布在很大程度上是连续的，因为车辆是由个人驾驶的，而且存在实际的速度差异。不过，我们可以从 "群 "的角度来理解密度，即密度的多重聚集。当高密度波束与低密度波束相遇时（由于速度与密度之间存在反比关系），"一束连续波可以凝聚成不连续波或'冲击波'。当车辆进入这种状态时，其平均速度会迅速大幅降低"（Lighthill 和 Whitham）。(莱特希尔和惠瑟姆 1955 年，第 43 页）。
It is possible to find the speed of a shockwave, and therefore its theoretical effects on traffic moving through it, using as a basis the flow density curve. Figure 2.2 below demonstrates this construction. 以流量密度曲线为基础，可以求出冲击波的速度，从而求出冲击波对通过冲击波的交通的理论影响。下图 2.2 展示了这种构造。
Figure 2.2 - Use of Flow Concentration Curve to Predict Local Conditions near Shockwave 图 2.2 - 利用水流浓度曲线预测冲击波附近的当地条件
Source: Lighthill and Whitham, 1955 资料来源莱特希尔和惠瑟姆，1955 年
Two traffic conditions, flow density A and flow density B are marked on the left hand graph, the speed of travel of these conditions is drawn parallel to the tangents of each point. The conditions meet at the heavy line on the right hand graph, which is the resulting shockwave travelling at the speed indicated by its slope. As expected, behind the shockwave density is less and the traffic travels faster. The shockwave therefore travels at an intermediate speed and can be drawn as the chord on the left-hand graph. 左图标出了两种交通状况，流量密度 A 和流量密度 B，这两种状况的行驶速度平行于每个点的切线。这两种交通状况在右图的重线处交汇，即由此产生的冲击波以其斜率所示的速度前进。不出所料，冲击波后面的密度较小，车流速度较快。因此，冲击波以中间速度前进，可在左侧图形中画出弦线 。
Note that from the left-hand graph of Figure 2.2 shockwaves can propagate spatially in three ways: forwards, where the velocity of the shockwave is positive; stationary, where the cord has a slope of 0 ; and backwards, where the velocity of the shockwave is negative. 请注意，从图 2.2 的左侧图形中可以看出，冲击波在空间上有三种传播方式：向前传播，冲击波的速度为正；静止传播，冲击波的斜率为 0；向后传播，冲击波的速度为负。
2.3.2 Richards 2.3.2 理查兹
Richards (1956) writing at the same time as Lighthill and Whitham drew on the established theory of fluid dynamics to explain traffic shockwaves; "as is well known in fluid mechanics, the end result of one volume of fluid overtaking another must be the formation of a shockwave. This is simply a density discontinuity, which travels through the fluid in a manner that is well determined by the laws of motion." (Richards 1956 pg. 153) However, traffic is unlike fluid in that there is a direct relation between density and velocity. This is the complicating factor which Richards and Lighthill and Whitham concern themselves with. 理查兹（1956 年）与莱特希尔和惠瑟姆同时撰文，利用流体力学的既定理论来解释交通冲击波："众所周知，流体力学中，一个体积的流体超越另一个体积的流体，其最终结果必然是形成冲击波。这只是密度的不连续性，它在流体中的传播方式完全由运动规律决定"。(理查兹，1956 年，第 153 页）然而，交通与流体不同，密度与速度之间存在直接关系。这就是理查兹、莱特希尔和惠瑟姆所关注的复杂因素。
To be sure, Richard's theory of shockwaves can be considered to be essentially identical to Lighthill and Witham's (Pipes 1964), but simply represented in another manner. Calculating the propagation of shockwaves through time and space using 'shearing equations', Richards demonstrates his theory with the diagram Figure 2.3 below: 可以肯定的是，理查德的冲击波理论与莱特希尔和威瑟姆的理论（Pipes，1964 年）基本相同，只是换了一种表达方式而已。理查德使用 "剪切方程 "计算冲击波在时间和空间中的传播，并用下图 2.3 展示了他的理论：
Figure 2.3 - Traffic Queue Build up and Dispersion 图 2.3 - 交通队列的集结与分散
Where is fractional Density, is distance A is a nominal fractional density of maximum velocity Source: Richards, 1956 其中 为分数密度， 为距离 A 为 最大速度的标称分数密度 来源：Richards 1956：理查兹，1956 年
A density discontinuity must have its origins somewhere, and Richards describes a disruption whose effects travel with time and space. At the disruption taking place at Time 0 and Distance 0, the fractional density behind the disruption increases from a (1/2) to 1 . At this time the queue extends back as far as -ac, which is the speed multiplied by the duration of the disruption multiplied by the fractional density a. In the succeeding time period (represented by the dotted line), the back of the queue propagates backwards, while traffic moving off from the point of disruption , do so at the pre-disruption fractional density, causing a starting shockwave represented by the diagonal dotted line. The stopping shockwave travels at -ac, while the starting shockwave travels at , so we would expect the starting shockwave to overtake the stopping shockwave upstream of the disturbance. 密度的不连续性必须起源于某个地方，理查兹描述了一种中断，它的影响会随着时间和空间的变化而变化。当中断发生在时间 0 和距离 0 时，中断后的分数密度从 a (1/2) 增加到 1。此时，队列向后延伸至 -ac，即速度乘以中断持续时间再乘以分数密度 a。在随后的时间段内（以虚线表示），队列向后传播，而从中断点 出发的车流则以中断前的分数密度移动，从而产生以对角虚线表示的起始冲击波。停止冲击波的传播速度为 -ac，而起始冲击波的传播速度为 ，因此我们预计起始冲击波会在干扰上游超过停止冲击波。
2.3.3 Second Order 2.3.3 二阶
The models proposed by Lighthill & Whitham, and Richards [LWR Models] portray shockwaves accurately at the macroscopic level, but only in cases where the flow is not highly dynamic (significant periods of stop and go traffic), and therefore not on the approach to busy intersections. They are therefore not equipped to model shockwaves at the venue of this study, the unsignalised roundabout. This is because the time in which one shock wave is generated may be very close indeed to the time of another, and the inherent approximation of these shockwaves at the same time produces large errors in the modelling of the movement of traffic (Gazis 2002). It is also flawed for light traffic where overtaking is possible because it does not recognize that there is a distribution of desired velocities across vehicles as well as a variation of the desired velocity for each vehicle. (Daganzo 1994). The velocity distribution across vehicles tends to have the effect in reality that platoons of vehicles disperse over time - this aspect of traffic flow is missing from the LWR model. For these reasons the models have been deemed suboptimal and have been advanced by subsequent studies. Lighthill & Whitham 和 Richards 提出的模型[LWR 模型]在宏观层面上准确地描述了冲击波，但只适用于车流动态性不强的情况（车流停停走走的时间较长），因此不适用于繁忙交叉口的进路。因此，它们不具备在本研究的地点（无信号环岛）模拟冲击波的能力。这是因为一个冲击波产生的时间可能与另一个冲击波产生的时间非常接近，而同时产生这些冲击波的固有近似方法会在交通运动建模中产生很大误差（Gazis，2002 年）。对于有可能超车的轻度交通，这种方法也存在缺陷，因为它没有认识到车辆之间的期望速度分布以及每辆车期望速度的变化。(Daganzo 1994）。在现实中，车辆间的速度分布往往会产生车辆排随时间而分散的效果，而 LWR 模型却没有考虑到交通流的这一方面。由于这些原因，这些模型被认为是次优的，并在随后的研究中得到了改进。
The LWR model was advanced by Payne (1971) into a 'second order model' by the inclusion of an additional differential equation that describes the dynamics of the ranges of velocity within all traffic states. The second order models are still macroscopic, and continue the analogy of fluid-dynamics, 佩恩（Payne，1971 年）将 LWR 模型提升为 "二阶模型"，增加了一个微分方程来描述所有交通状态下的速度范围动态。二阶模型仍然是宏观模型，继续沿用流体力学的类比方法、
but the assumption that vehicles instantaneously react to changes in density is removed, and replaced by a more realistic assumption of delay reaction. As such, the effect of the second differential equation (comprising this 'diffusion term') is to smooth shockwaves by altering particle reactions to more accurately reflect driver reactions. This has the theoretically advantageous result that vehicle velocity does not only depend on the local aggregate densities, but also on the gradient of the density through space; that is, drivers drive more slowly when density increases in the driving direction (Nagel 2003). 但取消了车辆对密度变化立即做出反应的假设，代之以更现实的延迟反应假设。因此，第二个微分方程（包括 "扩散项"）的作用是通过改变颗粒反应来平滑冲击波，从而更准确地反映驾驶员的反应。这在理论上有一个有利的结果，即车辆速度不仅取决于局部的聚集密度，还取决于空间的密度梯度；也就是说，当行驶方向上的密度增加时，驾驶员的驾驶速度会更慢（Nagel，2003 年）。
In 1994 Daganzo expanded on the improvements of Payne with a practical advance of the spatial monitoring of traffic. Instead of picturing a fluid traffic travelling along a single container, a series of cells pass traffic (and traffic densities) along the roadway. This method automatically generates density discontinuities in appropriate places while cutting down on extraneous calculations associated with the original LWR theorem. 1994 年，达甘佐在佩恩改进的基础上，在交通空间监测方面取得了实际进展。他不再描绘沿着单个容器行驶的流动交通，而是一系列单元格沿着道路传递交通（和交通密度）。这种方法可以在适当的地方自动产生密度不连续，同时减少与最初的 LWR 定理相关的无关计算。
Despite Daganzo's cellular approach, a full analysis of traffic flow and shockwaves using the second order models requires a large number of calculations. There are also drawbacks associated with the results of those calculations. One problem associated with the model is that drivers respond to waves of higher density that are downstream of them and moving at a higher speed (Aw & Rascle 2000). Another point of criticism is that the diffusion term predicts that vehicles at the back of a queue will have a negative velocity and so drive backwards. We can reasonably assume that both of these predictions are unrealistic and that the advances that the second order models attempt to offer over the LWR models contain critical flaws. Furthermore, the result of the sustained analogy with fluid dynamics is that vehicles respond to changes in density behind them (De Castillo, Pintado & Benitez 1993), again this can be considered an unrealistic proposal. 尽管达甘佐采用了单元方法，但使用二阶模型对交通流和冲击波进行全面分析需要进行大量计算。这些计算结果也存在缺陷。与该模型相关的一个问题是，驾驶员会对位于其下游并以更高车速行驶的高密度冲击波做出反应（Aw 和 Rascle，2000 年）。另一个受到批评的问题是，扩散项预测排在队列后面的车辆会有负速度，因此会向后行驶。我们可以合理地认为，这两种预测都是不现实的，而且二阶模型试图提供的相对于 LWR 模型的进步包含着关键的缺陷。此外，与流体动力学持续类比的结果是，车辆会对其后面的密度变化做出反应（De Castillo、Pintado 和 Benitez，1993 年），这同样可以被认为是一个不切实际的建议。
2.3.4 Second Order 'Resurrection' 2.3.4 二阶 "复活
The faults of the second order models outlined above have been corrected by scholars who simply derive the two differential equations differently. Aw and Rascle (2000) carried out a heuristic repair of the Payne (1971) model by reworking its mathematics with reference to a list of parameters. The parameters all refer either to hyperbolicity (wherein if values are specified for time then outputs can be generated for all time) and the Riemann Problem (relating to conservation of gases in shockwaves). It is considered that the paper removes all the inconsistencies of the Payne model, but that computation time is greatly increased. 上述二阶模型的缺陷已被学者们纠正，他们只是以不同的方式推导出两个微分方程。Aw 和 Rascle（2000 年）对 Payne（1971 年）模型进行了启发式修复，参照一系列参数重新编制了数学模型。所有参数都涉及双曲线性（如果为时间 指定数值，则可以产生所有时间的输出）和黎曼问题（与冲击波中的气体守恒有关）。本文认为消除了佩恩模型的所有不一致之处，但计算时间大大增加。
In Klar and Wegener (1996) it is shown that the standard model of the second order is unable to deal with inhomogeneous traffic. The models themselves, such as those developed by Paveri-Fontana (1979) and Prigogine and Herman (1971), incorporate a heuristically defined relation term which is equivalent to the diffusion term described above. Klar and Wegener find that the models show an improvement over the LWR pure fluid dynamic approach only in simulation of homogenous traffic state. Where flow is inhomogeneous, there is no mechanism in the model's equations to allow for density discontinuities, or shock waves, to propagate backwards along a flow of traffic. The Klar family of models (including a Fluid Dynamic version) all utilize a standard kinetic approach to some degree, drawing on an Enskog-type correction term derived from the Enskog theory of dense gases. Enskog's kinetic theory of gases consists in the introduction of corrections that "account for the fact that for dense gases the molecular diameter is no longer small compared with the average intermolecular distance" (Jakobsen 2008 pg. 319). Klar 和 Wegener（1996）的研究表明，二阶标准模型无法处理不均匀交通。Paveri-Fontana (1979)和 Prigogine 与 Herman (1971)等人开发的模型本身包含了一个启发式定义的关系项，相当于上述扩散项。Klar 和 Wegener 发现，只有在模拟同质交通状态时，这些模型才比 LWR 纯流体动力学方法有所改进。在流量不均匀的情况下，模型方程中没有允许密度不连续或冲击波沿车流向后传播的机制。克拉模型系列（包括流体动力学版本）都在一定程度上采用了标准动力学方法，借鉴了恩斯科格致密气体理论中的恩斯科格型修正项。恩斯科格的气体动力学理论包括引入修正项，以 "说明对于致密气体，分子直径与平均分子间距相比不再小"（Jakobsen，2008 年，第 319 页）。
The Klar model's architecture is arranged so that the fluid-dynamic model is derived from a gaskinetic model, which is in turn derived from a microscopic model. The fault here is not framed in terms of accuracy but in terms of practicality. Bellomo and Delitala (2002) observed that under some conditions the Klar system of calculations is not closed, and that therefore one may obtain 'an infinite hierarchy of equations'. In terms of the content of the model, the dynamics of vehicles interactions are determined only by pairs of headways, with those headways having a threshold beyond which vehicles are totally disinterested in one another. This is considered to be a rather restricted view of real driver behaviour. 克拉模型的结构安排是这样的：流体动力模型从加斯金模型中导出，而加斯金模型又从微观模型中导出。这里的问题不在于准确性，而在于实用性。Bellomo 和 Delitala（2002 年）指出，在某些条件下，Klar 计算系统并不是封闭的，因此可能会出现 "无限层级的方程"。就模型的内容而言，车辆相互作用的动态仅由成双成对的车头决定，这些车头有一个临界值，超过这个临界值，车辆就对彼此完全不感兴趣了。这被认为是对真实驾驶员行为的一种相当局限的看法。
2.4 Car Following Models 2.4 汽车跟踪模型
Shockwaves can be generated with varying degrees of plausibility using macroscopic models of traffic derived from general theories of the behaviour of fluids and gas. An alternative approach to traffic flow modelling, the car following approach, attempts to describe traffic in a more 'bottom-up' manner. Car following models begin by examining how a single car reacts to the car(s) ahead and then applies these behavioural parameters to a mass of cars. Each unit within a car following model is therefore modelled uniquely. There is generally far less description of shockwaves or any other multi-car phenomena in this school of literature. This does not mean that car following models are incapable of producing shockwaves, but merely that such phenomena would be a result of the models, as opposed to a focus in their production. It is therefore pertinent to discuss the basic premises of car following models and how they are constructed, before describing their production of shockwaves. 根据流体和气体行为的一般理论推导出的宏观交通模型可以产生不同程度的冲击波。交通流建模的另一种方法，即汽车跟随法，试图以更 "自下而上 "的方式描述交通。汽车跟随模型首先研究单辆汽车如何对前方汽车做出反应，然后将这些行为参数应用于大量汽车。因此，跟车模型中的每个单元都是单独建模的。在这一学派的文献中，对冲击波或任何其他多车现象的描述通常要少得多。这并不是说汽车跟随模型不能产生冲击波，而只是说这种现象是模型的结果，而不是产生冲击波的重点。因此，在描述汽车追尾模型产生冲击波之前，有必要讨论一下汽车追尾模型的基本前提及其构建方式。
2.4.1 Gazis Herman Rothery Models 2.4.1 加齐斯-赫尔曼-罗瑟里模型
Chandler, Hermann and Montrol (1958) [CHM] build their conception of car following on intuition and first principles - cars will attempt to follow the vehicle ahead, assuming no over-taking or interference from other vehicles, at a fixed distance headway. The reason for this is that drivers wish to avoid accidents. Where there are no other vehicles to follow, the driver will attempt to maintain a constant speed, but that attempt is foiled by the effects of road geometry. This is called acceleration or deceleration noise. In the instant of acceleration/deceleration noise the first vehicle will transfer an increased or decreased headway to the car following behind. Therein are the makings of "the criteria for the growth or decay of a disturbance" (Chandler Hermann and Montrol 1958 pg.7). The key factor in such a growth would be a driver's acceleration proportional to the change in relative speed, subject to delayed response factors. 钱德勒、赫尔曼和蒙特罗尔（1958 年）[CHM] 根据直觉和第一原理提出了汽车跟车的概念--假设没有超车或其他车辆的干扰，汽车会试图以固定的距离跟在前车后面。这样做的原因是驾驶员希望避免事故。在没有其他车辆跟随的情况下，驾驶员会试图保持一个恒定的速度，但这一尝试会受到道路几何形状的影响。这就是所谓的加速或减速噪声。在加速/减速噪音产生的瞬间，前车会将增加或减少的车头速度传递给后车。这就是 "干扰增长或衰减的标准"（Chandler Hermann 和 Montrol，1958 年，第 7 页）。这种增长的关键因素是驾驶员的加速度与相对速度的变化成正比，并受到延迟反应因素的影响。
Gazis, Herman and Potts (1959) [GHP] seek to define a relationship between the microscopic car following models with the empirically confirmed flow/density graphs outlined in Section 2.1 above. They do so by modifying the CHM approach with a sensitivity factor which is inversely proportional to headway. Hermann and Potts (1959) calibrated the model using wire-linked vehicles through tunnels in the New York area, finding that the data produced a good fit with the models suppositions. Gazis、Herman 和 Potts（1959 年）[GHP] 试图定义微观汽车跟随模型与上文第 2.1 节所述经验证实的流量/密度图之间的关系。为此，他们对 CHM 方法进行了修改，增加了一个与车速成反比的敏感系数。Hermann 和 Potts（1959 年）使用通过纽约地区隧道的线控车辆对模型进行了校准，发现数据与模型假设非常吻合。
Gazis Herman and Rothery (1961) [GHR], acknowledging the increasing numbers of competing car following models emerging at the time, present a reappraisal of the theories in general and offer specific additional results. They do so with the express dichotomy that car following models for high density scenarios should differ from models for low density models, where car interactions are statistically very low. Gazis Herman 和 Rothery（1961 年）[GHR] 认识到当时出现了越来越多相互竞争的汽车跟随模型，对这些理论进行了重新评估，并提供了具体的补充结果。他们提出了一个明确的二分法，即高密度情况下的汽车跟随模型应不同于低密度模型，因为在低密度模型中，汽车之间的相互作用在统计学上非常低。
Using data from tunnels in New York and also from the General Motors test track, a pattern of variation of sensitivity with respect to headways and speed was established, and as in GHP, the relationship of the 'phenomenological' macroscopic densities are compared with the above micro level vehicle behaviour models. Data from the test track was used to define the constant terms of the car following model. Computation of the data showed the same constant terms repeatedly, giving the GHR approach the landmark status that it enjoys today. 利用来自纽约隧道和通用汽车公司试验场的数据，确定了灵敏度随车头和车速变化的模式，并将 "现象学 "宏观密度与上述微观车辆行为模型进行了比较。来自测试轨道的数据被用来定义汽车跟随模型的常数项。对数据的计算重复显示了相同的常数项，从而使 GHR 方法获得了今天的里程碑地位。
In the 15 years that followed the GHR work, a number of papers emerged proposing different constant factors to be built into the model. The propagation of car following models in the GHR frame and the repeated efforts to finely calibrate them is notable for its unintended consequences. With no one model being a clear winner and because of the large number of contradictory findings as the correct constant factors, the models are rarely used. The reason for the contradictory findings has been well described by Rockwell and Treiterer (1966) who propose that microscopic behaviour is likely to alter with flow and density i.e. the two-model approach of GHR is far too few. And secondly "many of the empirical investigations have taken place at low speeds or in extreme stop start conditions, which may not reflect more general car-following behaviour" (Brackstone 1999 pg. 185). 在 GHR 工作之后的 15 年中，出现了许多论文，提出在模型中加入不同的常数因子。在 GHR 框架下，汽车模型的传播以及对这些模型进行精细校准的反复努力产生了意想不到的后果。由于没有一个模型是明显的赢家，而且在正确的常数因子方面存在大量相互矛盾的结论，因此这些模型很少被使用。罗克韦尔和 Treiterer（1966 年）已经很好地描述了出现矛盾结论的原因，他们提出微观行为可能会随流量和密度的变化而改变，即 GHR 的双模型方法太少。其次，"许多经验调查都是在低速或极端的停车起步条件下进行的，这可能无法反映更普遍的汽车追随行为"（Brackstone，1999 年，第 185 页）。
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2.4.2 Safety Distance or Collision Avoidance Models (CA Models) 2.4.2 安全距离或避免碰撞模型（CA 模型）
The safety distance school of models was originally formulated by Kometani and Sasaki . The CA models differ from the GHR models in that they do not seek to predict how one car reacts to the acceleration or deceleration of the car ahead, but rather to predict safe following distances, and less safe following distances "within which a collision would be unavoidable were the lead vehicle to display unpredictable driving behaviour" (Brackstone 1999 pg. 186). Repeated calibration of the model using test track data found differing reaction time factors and indeed differing following distances, all depending on driver characteristics. It is these safety distances and reaction times which are used to determine how each vehicle behaves. The determination takes place within a probability distribution of driver characteristics. 安全距离模型流派最初由 Kometani 和 Sasaki 提出 。CA 模型与 GHR 模型的不同之处在于，它们并不试图预测一辆车对前车加速或减速的反应，而是预测安全跟车距离，以及 "如果前车表现出不可预测的驾驶行为，碰撞将不可避免 "的较小安全跟车距离（Brackstone，1999 年，第 186 页）。利用试车场数据对模型进行反复校准后发现，反应时间系数和跟车距离都有所不同，这一切都取决于驾驶员的特点。正是这些安全距离和反应时间被用来确定每辆车的行为方式。这种确定是在驾驶员特征的概率分布范围内进行的。
Gipps (1981) added to the CA model described above additional safety considerations that drivers can reasonably be assumed to employ, but that were not in the Kometani and Sasaki formula. For instance, drivers allow an additional safety reaction time and a term representing the varied braking rates in different scenarios. This was done to achieve the criteria that all traffic models should mimic the behaviour of real traffic and real drivers. Without such realism, he contends that the prior CA models would have vehicles driving at their preferred speed at an obstacle, before employing their maximum possible deceleration rate, i.e. screeching to a halt. It is considered that the model satisfies the criteria of realism since its tenets correspond to assumed driving characteristics. Though there is no calibration offered within the paper, it is validated using realistic values and found to produce results commensurate to a satisfactory degree of realism. 吉普斯（1981 年）在上述 CA 模型的基础上，增加了驾驶员可以合理假定采用的、但在 Kometani 和 Sasaki 公式中没有的安全考虑因素。例如，驾驶员允许额外的安全反应时间和一个代表不同情况下不同制动率的项。这样做是为了达到所有交通模型都应模仿真实交通和真实驾驶员行为的标准。他认为，如果没有这种真实感，先前的 CA 模型就会使车辆在遇到障碍物时以自己喜欢的速度行驶，然后再采用可能的最大减速率，即急刹车。他认为，该模型符合现实主义标准，因为其原理与假定的驾驶特性相符。虽然本文没有提供校准，但使用现实值对其进行了验证，发现其产生的结果与令人满意的现实程度相符。
2.4.3 Optimal Velocity Model 2.4.3 最佳速度模型
The optimal velocity approach to car following frames driver decisions in terms of their attempts to follow a preferred, or optimal, velocity. Following drivers accelerate and decelerate on the divergence between this optimal velocity and the constraints of other traffic, acceleration noise, and geometric concerns. The model, developed by Bando, Hasebe, Nakayama, Shibata and Sugiyama (1995), predicts that all vehicles drive at the same speed in low density conditions, which is to say that car followers are in effect speed-takers. But when traffic flow begins to develop higher densities the homogenous solution becomes unstable and eventually breaks down into a system of multiple jams, each separated by brief pockets of free flowing traffic. 最佳速度跟车方法将驾驶员的决策定格在他们试图跟车的首选速度或最佳速度上。跟车驾驶员根据最佳速度与其他交通限制、加速噪声和几何因素之间的偏差进行加速和减速。根据 Bando、Hasebe、Nakayama、Shibata 和 Sugiyama（1995 年）建立的模型预测，在低密度条件下，所有车辆都以相同的速度行驶，也就是说，跟车司机实际上是速度接受者。但是，当车流密度开始增大时，同质解决方案就会变得不稳定，并最终分解成一个由多个拥堵点组成的系统，每个拥堵点之间都有短暂的畅通车流。
The inclusion of an optimal velocity term in a model determines that drivers only respond to space headways, and ignore totally the speeds of other vehicles. This is considered to be a gross simplification which has no basis in reality. For that reason Wagner (2010) states that the theory is, at the microscopic level at least, seriously flawed. Practically speaking, optimal velocity models have the fairly pronounced drawback that they produce crashes. The crashes are not born of an attempt to model road safety, but rather, there is no mechanism that prevents vehicle's 'relaxation periods' from overlapping and crashing. So crashes occur very frequently. The models further divert from reality with unrealistically high value accelerations and decelerations. 在模型中加入最佳速度项，就意味着驾驶员只对车距做出反应，而完全忽略了其他车辆的速度。这被认为是一种严重的简化，在现实中毫无根据。因此，瓦格纳（2010）指出，该理论至少在微观层面上存在严重缺陷。实际上，最佳速度模型有一个相当明显的缺点，那就是会产生碰撞。这些撞车事故并不是因为试图建立道路安全模型，而是因为没有任何机制可以防止车辆的 "松弛期 "重叠和撞车。因此，撞车事故频频发生。由于加速度和减速度的数值过高，模型进一步偏离了现实。
2.4.4 Psychophysical or Action Point Models 2.4.4 心理物理或行动点模型
Driver decisions in car following scenarios are stimulated only by those dimensions of a changing environment that are perceptible to him, and are only acted upon through a mechanism which distorts his driving intentions. In considering the car following scenario as stated, Michaels (1963) investigates three conditions in which the human role can be understood. The first is overtaking, the second is steady-state traffic and the third is response to acceleration and deceleration. It is the third of these conditions which is most relevant to the propagation of shockwaves. 在跟车情景中，驾驶员的决策仅受到其可感知的环境变化因素的刺激，并且仅通过一种扭曲其驾驶意图的机制发挥作用。Michaels （1963 年）在考虑上述跟车情景时，研究了三种可以理解人类作用的情况。第一种是超车，第二种是稳态交通，第三种是对加速和减速的反应。其中第三种情况与冲击波的传播最为相关。
The steady state is disturbed by significant acceleration or deceleration to which following drivers must respond. Clearly, as with all human responses, there is a time lag between the stimulus and 稳定状态会受到明显加速或减速的干扰，而跟车驾驶员必须对此做出反应。很明显，与人类的所有反应一样，刺激和反应之间存在时间差。
the response. The lag is determined in large part by the delay from the onset of acceleration to the time when the velocity reaches a threshold. The velocity threshold derives from the detection of a change in the visual angles between the lead and the following vehicles. There is therefore a distance at which a change in velocity is 'just noticeable'. Since there is in Michaels' conception a "Just Noticeable Distance" there is a certain degree of acceleration which goes unnoticed by following drivers. The other delay factor is between the detection and the response of the vehicle, this delay is mechanical in nature. Once the driver's decision is played out on the road, drivers will choose to reverse their acceleration/deceleration "until they can no longer perceive any relative velocity, and provided the threshold is not then re-exceeded, will base all their actions on whether they can then perceive any changes in spacing" (Brackstone 1999 pg. 190). 反应。滞后在很大程度上取决于从开始加速到速度达到阈值的延迟时间。速度阈值来自于对前车和后车之间视觉角度变化的检测。因此，速度变化 "刚刚能被注意到 "的距离是存在的。由于在迈克尔斯的概念中存在一个 "刚刚可以注意到的距离"，因此会有一定程度的加速，而跟车驾驶员不会注意到。另一个延迟因素存在于车辆的检测和反应之间，这种延迟是机械性的。一旦驾驶员的决定在道路上得到执行，驾驶员就会选择反向加速/减速，"直到他们无法再感知到任何相对速度为止，只要不再超过阈值，他们的所有行动都将以是否能感知到间距的任何变化为基础"（Brackstone，1999 年，第 190 页）。
Experiments carried out by Evans and Rothery (1973) sought to measure the supposed perception thresholds that Michaels had proposed. Subjects in a series of road tests were driven in varying car following scenarios and asked whether or not they could perceive changes in relative motion. There were two major conclusions from the study which were to feed into the makeup of Action Point models generally; the dominant cue used to establish the onset of relative motion is the average value of relative speed divided by the distance headway, simply put, drivers are more able to observe deceleration than they are acceleration; and, drivers are very sensitive to relative motion. This last conclusion indicates that traffic will unlikely be smoothed by providing drivers with information on likely acceleration and deceleration waves forming ahead. 埃文斯和罗瑟里（1973 年）进行的实验试图测量迈克尔斯提出的所谓感知阈值。在一系列道路测试中，受试者在不同的汽车跟随场景中行驶，并询问他们是否能感知到相对运动的变化。这项研究得出了两个主要结论，这两个结论将普遍应用于行动点模型的构成；用于确定相对运动开始的主要线索是相对速度的平均值除以前进距离，简单地说，驾驶员观察减速的能力比观察加速的能力更强；驾驶员对相对运动非常敏感。最后一个结论表明，向驾驶员提供前方可能形成的加速波和减速波的信息，不太可能使交通变得顺畅。
2.4.5 Model Assessment 2.4.5 模型评估
There is a very large number of competing models and theories of traffic flow. It is understandable that so many scholars would be so keen to explain traffic flow, since it takes place in a public setting and thus it is feasible that authorities would wish to manage it. 关于交通流，有大量相互竞争的模式和理论。众多学者热衷于解释交通流是可以理解的，因为交通流发生在公共环境中，因此当局希望对其进行管理是可行的。
One assumes that there is tacit acknowledgement that such models will never reach perfection that they will never perfectly reflect traffic as it flows in reality. It follows then that there is a point at which additional work begins to yield reduced returns, which is to say, the field experiences diminishing returns. This paper is by no means bold enough to suggest that that point has been passed, but doubtless the array of competing theories would appear far more useful if it were accompanied by a similarly detailed and numerous body of comparative papers. There is a relative paucity of studies which compare the many theories against one another theoretically and/or empirically in terms of their usefulness. 我们假定，人们默认这些模型永远不会达到完美，它们永远不会完美地反映现实中的交通流量。因此，在某一点上，额外的工作开始产生较少的回报，也就是说，该领域的回报会递减。本文绝不是大胆地认为这一点已经过去，但毫无疑问，如果有同样详细和大量的比较性论文，各种相互竞争的理论就会显得有用得多。从理论和/或实证角度对众多理论的实用性进行比较的研究相对较少。
One such comparative paper was produced by Wagner in 2010. The study carries out a data driven comparison of 6 divergent approaches to traffic flow theory. Of this 6 , the ones pertinent to the theories previously recounted in this paper are as follows: 瓦格纳（Wagner）于 2010 年撰写了一篇此类比较论文。该研究以数据为驱动，对 6 种不同的交通流理论方法进行了比较。在这 6 种方法中，与本文之前叙述的理论相关的方法如下：
Daganzo's 1994 second order model with cellular transmission CT - as discussed in Section 2.3.3 Second Order2.3.3 Second Order 2.3.3 二阶2.3.3 二阶
Aw and Rascle 2000 model AW - as discussed in Section 2.3.4 Second Order 'Resurrection' Aw 和 Rascle 2000 模型 AW - 如第 2.3.4 节所述 二阶 "复活
Stephan Krauss' 1997 cellular and collision avoidance model, based on Gipps' 1981 Colission Avoidance model SK - as discussed in Section 2.4.2 Safety Distance or Collision Avoidance Models (CA Models) 斯蒂芬-克劳斯（Stephan Krauss）1997 年基于吉普斯（Gipps）1981 年避撞模型 SK 的蜂窝和避撞模型--如第 2.4.2 节安全距离或避撞模型（CA 模型）所述
The Mitsim Model, introduced by Ahmed in 1999. The model employees three different regimes depending on space headways; low headway; medium headway - (based on the GHR model of 1961); high headway Mitsim 模型，由 Ahmed 于 1999 年提出。该模型根据空间净空的不同提供三种不同的模式：低净空；中净空（基于 1961 年的 GHR 模式）；高净空。
Each model was calibrated to represent an uninterrupted section of 5 lane freeway in California. Observed data from the freeway was input into each model and a process of optimisation was carried out, wherein parameters were fit to the scenario, thus validating the models. The speeds generated by the models over a whole day were then compared with the speeds as measured by inductive loop, and the root mean square (R.M.S) error of difference was calculated. The results of this process are shown in Figure 2.4 below. 每个模型都经过校准，以代表加利福尼亚州一段不间断的 5 车道高速公路。高速公路上的观测数据被输入到每个模型中，并进行了优化处理，其中的参数被拟合到情景中，从而验证了模型。然后，将模型生成的全天车速与感应圈测得的车速进行比较，并计算出差异的均方根误差 (R.M.S)。这一过程的结果如下图 2.4 所示。
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As the authors admit, there is a tendency for models carrying a larger number of parameters to suffer high R.M.S errors due to failures in the optimisation process. This is intuitive since if there are more criteria to fit in one model than in another then it follows that the first model is likely to require less work in fitting. This explains the weakness of the Mitsim model, which has many more parameters than the others tested. 正如作者所承认的，由于优化过程中的失误，参数数量较多的模型往往会出现较高的 R.M.S 误差。这一点很直观，因为如果一个模型比另一个模型有更多的拟合标准，那么第一个模型的拟合工作就可能更少。这就解释了 Mitsim 模型的弱点，因为该模型比其他测试模型有更多的参数。
The CT model is the weakest of the LWR School, with the modifications of AR appearing to have achieved a degree of extra accuracy. Overall though, the Mitsim parameter issue withstanding, there does not appear to be a great discrepancy of error rates among the models tested. This would appear to suggest that with adequate optimisation any of modelling approaches can yield comparably adequate results. CT 模型是低纬度流派中最弱的，而 AR 模型的修改似乎在一定程度上提高了精确度。总体而言，尽管存在 Mitsim 参数问题，但所测试的模型之间的误差率似乎并无太大差异。这似乎表明，只要进行适当的优化，任何建模方法都可以产生相当充分的结果。
2.4.6 Car Following Models and Shockwaves 2.4.6 汽车追尾模型和冲击波
The car following models discussed above have added substantially to academia's understanding of traffic flow. The development of models to explain each vehicle's behaviour singularly was perhaps inevitable, given that units of traffic are at once cellular and human-driven. But it is important to stress that these models can and do still produce macroscopic phenomena. More specifically they are capable of producing the kind of shockwaves that LWR theorised. 上文讨论的汽车跟随模型极大地丰富了学术界对交通流的理解。鉴于交通单位既是细胞，又是由人驱动的，开发模型来单独解释每辆车的行为也许是不可避免的。但必须强调的是，这些模型仍然能够并确实产生宏观现象。更具体地说，它们能够产生 LWR 理论中的那种冲击波。
Newell (1961) interrogates a simple car following model and finds that it allows for the production of shockwaves. Using a car following model and a significant amount of mathematics, Newell derives the relationship described in Figure 2.5. 纽厄尔（1961 年）对一个简单的汽车跟随模型进行了研究，发现该模型允许产生冲击波。纽厄尔利用汽车追尾模型和大量数学计算，得出了图 2.5 所描述的关系。
Below are marked two points on the velocity-headway curve associated with the velocities (1-a)V and (1-b)V. The slope between these two points is equal to the change in headway divided by the change in velocity. The point at which this slope crosses the velocity axis is the velocity of the shock. It is expected that this shock results in each car experiencing identical changes in velocity as it predecessor, only later in time. 下面标出了速度-航程曲线上与速度 (1-a)V 和 (1-b)V 相关的两个点。这两点之间的斜率等于前进速度的变化除以速度的变化。该斜率与速度轴的交叉点即为冲击速度。预计冲击会导致每辆车的速度变化与其前一辆车相同，只是时间上更晚一些。
Figure 2.5 - Shock Velocity in a Car Following Model Source: Newell 1961 图 2.5 - 汽车追尾模型中的冲击速度 资料来源：纽厄尔，1961 年：纽厄尔 1961
Figure 2.6 - Deceleratory Shock in a Car Following Model 图 2.6 - 跟车模型中的减速冲击
Source: Newell 1961 资料来源资料来源：Newell 1961
Figure 2.6 above shows the velocity of a sequence of cars when the lead car suddenly decelerates from 0.5 to 0 , where is free flow velocity and vj is actual velocity of the vehicle . Vehicle j0 instantaneously stopping is represented by the vertical line at 0 on the axis where the axis is plotted as relative time. After vehicle stops, the vehicles behind all decelerate and apart from the first few cars where the shockwave is still forming, every car goes through the same deceleration profile, just offset by delta in time. 上图 2.6 显示了当领头车突然从 0.5 减速到 0 时一连串车辆的速度，其中 为自由流速度，vj 为车辆的实际速度 。车辆 j0 在 轴 0 处的垂直线表示车辆瞬间停止， 轴表示相对时间。车辆 停止后，后面的车辆都会减速，除了冲击波仍在形成的前几辆车外，每辆车都会经历相同的减速曲线，只是在时间上偏移了 delta。
Newell concludes that the car following model under examination presents the temporal and spatial progression of shocks that is central to the macroscopic theories, but with closer attention towards 纽厄尔的结论是，所研究的汽车追尾模型呈现了冲击在时间和空间上的发展过程，这是宏观理论的核心，但更关注的是
individual vehicles and their travel through those shocks, this includes their acceleration and deceleration. Thus "the model includes practically everything that has been contained in any older models for dense traffic flow and, in addition, allows one to investigate such things as the development of shocks, shock profiles, [...] and the spreading of an acceleration wave" (Newell 1961 pg. 267) The fact that shockwaves can be investigated using car following models as well as macroscopic models is of some significance to this paper, since it necessitates more choice in deciding how best to proceed with the experimental aspect of the project. 该模型可用于研究单个车辆及其通过这些冲击的情况，包括车辆的加速和减速。因此，"该模型实际上包含了密集交通流旧模型中包含的所有内容，此外，还可以研究冲击的发展、冲击剖面[......]以及加速波的扩散等问题"（Newell，1961 年，第 267 页）。冲击波可以使用汽车跟踪模型和宏观模型进行研究，这对本文具有一定的意义，因为在决定如何最好地进行项目的实验方面，需要有更多的选择。
2.5 Traffic Flow and Accidents 2.5 交通流量和事故
It has long been understood that accidents have many causes. It is rarely ever appropriate to suggest that an accident was the result of a single factor. Instead, road safety experts attempt to understand how a plurality of conditions can come together and cause an accident. This approach does not preclude the detailed study of factors in isolation however, and one affecting factor that scholars have attempted to understand in more detail is traffic flow. 人们早已明白，事故的发生有多种原因。认为事故是由单一因素造成的说法很少恰当。相反，道路安全专家试图了解多种情况如何共同导致事故的发生。然而，这种方法并不排除对单独因素的详细研究，学者们试图更详细地了解的一个影响因素是交通流量。
Studies of the relationship between traffic flow and accidents can generally be separated into two categories; those that use aggregate data such as AADT aligning statistically high flows with accident rates; and those that identify accidents along a route and attempt to find nearby crash precursors in the minutes prior to the accident. While the aggregate studies have yielded a plethora of interesting results and are methodologically secure (owing to their primarily statistical approach) there is understandably no discussion of shockwaves. Some of the studies have investigated different 'meso level' traffic conditions, such as 'congestion' or 'free flow', but are not able to go into sub categorical phenomena. 关于交通流量与事故之间关系的研究一般可分为两类：一类是使用总量数据（如平均日交通流量），将统计意义上的高流量与事故率联系起来；另一类是确定沿线的事故，并试图在事故发生前几分钟内找到附近的事故前兆。虽然总体研究得出了大量有趣的结果，而且在方法上也很可靠（由于主要采用统计方法），但没有对冲击波进行讨论，这是可以理解的。一些研究调查了不同的 "中间层 "交通状况，如 "拥堵 "或 "自由流动"，但无法深入研究细分现象。
Chang & Xiang (2003) studied flow and accidents at the 'meso level' and found that on motorways and arterial routes AADT is a significant positive factor on crash rate, as is peak-congestion conditions. During the off peak period however, there appears to be no statistical relationship between flow and accidents, suggesting that other contributory factors become more prominent under free flow conditions. Chang 和 Xiang（2003 年）在 "中观层面 "研究了流量和事故，发现在高速公路和干线公路上，AADT 与高峰拥堵条件一样，是影响事故率的重要积极因素。然而，在非高峰期，流量与事故之间似乎没有统计关系，这表明在自由流动条件下，其他促成因素变得更加突出。
Figure 2.7 - Accidents and RFC 图 2.7 - 事故和 RFC
Source: Zhou and Sisiopiku (1997) 来源：Zhou and Sisiopiku (1997)资料来源：Zhou 和 Sisiopiku (1997)
Zhou & Sisiopiku (1997) go somewhat further and treat congestion as fully continuous, investigating the relationship between the ratio of flow to capacity (RFC) and rate of accidents on a length of urban motorway. Their results are shown in Figure 2.7 above, where a U-like best-fit line is fit to the data points. The graph shows that in very low and very high RFC conditions - I.e. non congested and very congested periods respectively - we would predict high accident rates. It is considered that accidents taking place in free flow conditions are more likely to be reported as they take place at higher speeds. This may be the reason for higher number of accidents in non-congested conditions, and highlights one of the many challenges involved in analysing accident statistics. Zhou & Sisiopiku（1997 年）更进一步，将拥堵视为完全连续的，研究了 一段城市高速公路上流量与通行能力之比（RFC）与事故率之间的关系。他们的研究结果如上图 2.7 所示，图中的数据点与一条 U 型最佳拟合线相吻合。从图中可以看出，在极低和极高的 RFC 条件下，即分别在不拥堵和非常拥堵的时段，我们可以预测事故率会很高。我们认为，在自由流条件下发生的事故更有可能被报告，因为事故发生时车速较高。这可能是非拥堵状态下事故数量较高的原因，同时也凸显了事故统计分析所面临的诸多挑战之一。
Lord, Manar & Vizioli (2005) investigated the effect of both RFC and traffic density on accident rates and found that as density increases, crash frequency increases, before reaching a maximum and then decreasing again. This corroborates the findings presented in Figure 2.7 above. Lord、Manar 和 Vizioli（2005 年）研究了 RFC 和交通密度对事故率的影响，发现随着密度的增加，碰撞频率也会增加，在达到最大值之前会再次降低。这与上图 2.7 中的研究结果相吻合。
In each of these studies however the methodology was unable to isolate congestion and flow as factors separate from environmental conditions such as darkness and fatigue (Marchesini and Weijermars 2010) 然而，在这些研究中，每项研究的方法都无法将拥堵和流量作为独立于黑暗和疲劳等环境条件的因素（Marchesini 和 Weijermars，2010 年）。
2.5.2 Traffic Flow and Accidents - Disaggregate Studies 2.5.2 交通流量和交通事故--分类研究
The benefit of a disaggregate approach to accident analysis is that it captures short term phenomena that is missed by aggregate hourly and daily level approaches. Of course the added detail requires more data and more time, but even so a number of studies have emerged in recent years which have reasonably drawn conclusions on short term accident precursors. 对事故进行分类分析的好处在于，它可以捕捉到每小时和每天的综合分析方法所忽略的短期现象。当然，增加的细节需要更多的数据和更多的时间，但即便如此，近年来还是出现了一些研究，对短期事故前兆做出了合理的结论。
One such study produced by Oh, Oh, Ritchie and Chang (2001) evaluated crash likelihood on California motorways using traffic data recorded by loop detectors. Standard deviation of speed was used as an indicator of 'traffic disruption'. It was found that as speed variability increases so does the likelihood of traffic accidents. A study by Lee, Saccomano and Helinga (2002) obtained the same result with regards to speed variation by analysing loop data from urban motorways around Toronto. They further posited that relying on a single metric, as Oh, Oh, Ritchie and Chang did, is a questionable method of crash prediction since crashes occur as a result of multiple factors. An additional two statistically significant accident precursors were found to support this criticism. The full list of precursors found by Lee, Saccomano and Helinga is bulleted below. 由 Oh、Oh、Ritchie 和 Chang（2001 年）进行的一项此类研究利用环路探测器记录的交通数据评估了加利福尼亚州高速公路上发生交通事故的可能性。速度的标准偏差被用作 "交通中断 "的指标。结果发现，随着车速变化的增加，交通事故发生的可能性也在增加。Lee、Saccomano 和 Helinga（2002 年）通过分析多伦多周边城市高速公路的环路数据，得出了与速度变化相同的结果。他们进一步指出，像 Oh、Oh、Ritchie 和 Chang 所做的那样，依靠单一指标来预测交通事故的方法值得商榷，因为交通事故的发生是多种因素共同作用的结果。他们还发现了另外两个在统计学上具有重要意义的事故前兆，以支持这一批评。Lee、Saccomano 和 Helinga 发现的全部前兆列表如下。
The average variation of speed on each lane 每条车道上的平均速度变化
The average variation of speed differences across adjacent lanes 相邻车道速度差的平均变化量
Traffic density 交通密度
Though not a precursive factor in itself, it is nevertheless important to consider the impact of exposure. In their study it is described as "the product of daily traffic volume and the length of each road section" (Lee, Saccomano and Helinga 2002 pg.5). The effect of exposure is proportionate to the amount of exposure, which is a notable difference from the results of the aggregate studies shown above. 尽管这本身并不是一个前兆因素，但考虑暴露的影响也很重要。在他们的研究中，这种影响被描述为 "日交通量与各路段长度的乘积"（Lee、Saccomano 和 Helinga，2002 年，第 5 页）。暴露的影响与暴露量成正比，这与上述综合研究的结果明显不同。
A later study by the same authors (Lee, Helinga and Saccomano 2003) found that the average variation of speed differences across adjacent lanes has a negligible impact. This factor is therefore removed from the bulleted list above. The error was a result of an arbitrary definition of what time slice to look within for precursors. The succeeding study determined an appropriate time slice by finding the time at which the difference between precursor values are maximised for crash cases and non-crash cases. 同一作者后来进行的一项研究（Lee、Helinga 和 Saccomano，2003 年）发现，相邻车道之间的平均速度差异影响微乎其微。因此，该因素已从上表中删除。这一错误是由于任意定义了寻找前兆的时间片。后续研究通过找出碰撞案例和非碰撞案例中前兆值差异最大的时间，确定了合适的时间片。
The results of the study show that the average variation of speed differences across adjacent lanes is not as significant as initially proposed. It is replaced by average speed difference on upstream and downstream detectors (that is one detector either side of the accident location). The authors conclude that the speed differential between the detectors was significantly higher prior to crashes. The importance of this finding is that sharp transition of speed within the road section is a significant 研究结果表明，相邻车道的平均速度差变化并不像最初提出的那样显著。取而代之的是上下游探测器（即事故地点两侧的探测器）的平均速度差。作者的结论是，在发生碰撞事故之前，探测器之间的速度差明显较高。这一发现的重要性在于，路段内车速的急剧变化是造成事故的一个重要原因。
precursor of accidents, implying that the dynamics of a traffic queue, its emergence and dissipation, can be a key element in causing accidents. It may also imply that shockwaves increase accident rates. 这意味着交通队列的动态变化、其出现和消散可能是导致事故的关键因素。这也可能意味着冲击波会增加事故发生率。
There has been little study which explicitly deals with the relationship of shockwaves and accidents. This is perhaps due to the difficulty of gathering sufficient data so that shockwaves are able to be presented accurately at small time intervals and over long distances. Nevertheless, those studies which have identified a causal relationship do so convincingly. 很少有研究明确涉及冲击波与事故的关系。这可能是由于难以收集足够的数据，因此无法在较小的时间间隔和较远的距离上准确地呈现冲击波。然而，那些确定了因果关系的研究却令人信服。
Abdel & Aty & Pande (2006) studied the significant conditions in time and space for rear end collisions, or shunts, on motorways. It was found that the frequency of accidents altered when traffic speeds are varied through time and space, along a certain link and during a selected timeslice. Such variance in two dimensions may represent traffic shockwaves. Crashes were found to be more frequent at the back of a stopping shockwave than at the front of a starting shockwave. Abdel & Aty & Pande（2006 年）研究了高速公路上发生追尾碰撞或分流的重要时间和空间条件。研究发现，当交通速度在时间和空间上发生变化时，事故发生频率也会随之改变。这种两个维度的变化可能代表了交通冲击波。研究发现，在停止冲击波的后方发生交通事故的频率要高于在开始冲击波的前方。
Xu, Liu, Wang, Wang and Li (2012) investigated density discontinuities either side of a loop detector as part of a more general categorisation of traffic. The studies' 5 categories of traffic flow are listed as follows: Xu、Liu、Wang、Wang 和 Li（2012 年）研究了环形检测器两侧的密度不连续性，作为更普遍的交通分类的一部分。研究中的 5 种交通流分类如下：
Traffic state 1 exhibits a great difference in traffic occupancy between upstream and downstream loop detectors. 流量状态 1 显示上下游环路检测器之间流量占用率的巨大差异。
Traffic state 2 is characterized by quite homogeneous high occupancy traffic flow across four loop detectors. 流量状态 2 的特点是四个环路检测器的流量非常均匀，占用率很高。
In traffic state 3, traffic flow at four loop detectors are in the transition state between free flow and congested flow 在流量状态 3 中，四个环路检测器的流量处于自由流和拥堵流之间的过渡状态
The characteristics of traffic state 4 are opposite to that of traffic flow state 1. Traffic state 4 represents the situation in which the upstream traffic flow is in a free flow state while the downstream traffic flow is in a congested flow state. 交通流状态 4 的特征与交通流状态 1 相反。交通流状态 4 表示上游交通流处于自由流状态，而下游交通流处于拥堵流状态。
Traffic state 5 represents the condition in which both upstream and downstream freeway traffic is in free flow. 交通状态 5 表示高速公路上下游交通都处于自由流动状态。
Statistical analysis was carried out to test for a significant relationship between these categories and the occurrence of accidents. It was found that traffic state four had the greatest and most significant impact on accident rates; accidents were 7.11 times more likely to occur under the traffic state 4 conditions than under those of traffic state 5. It should be noted that although this study did not explicitly state that the density discontinuities were moving through space and time, it can reasonably be held that such phenomena is akin to the kind of shockwaves described in the sections above, if only because we know from first principles that density discontinuities are prone to having a sustainable speed and direction. 我们进行了统计分析，以检验这些类别与事故发生率之间是否存在显著关系。结果发现，交通状态 4 对事故发生率的影响最大，也最显著；交通状态 4 条件下发生事故的概率是交通状态 5 条件下的 7.11 倍。应该指出的是，虽然这项研究没有明确指出密度不连续是在空间和时间上移动的，但我们可以合理地认为，这种现象类似于上文所述的冲击波，这仅仅是因为我们从第一性原理中知道，密度不连续容易具有可持续的速度和方向。
Lee and Volpatti (2010) carried out a study which examines how shockwaves affect the likelihood of crashes on motorways. Using data from a motorway near Toronto, shockwaves were categorised by type, by comparing changes in density over time with reference to the capacity density. An example is given in Figure 2.8 below. There is a maximum of eight shockwave types that are functionally unique. Lee 和 Volpatti（2010 年）开展了一项研究，探讨冲击波如何影响高速公路上发生碰撞的可能性。他们利用多伦多附近一条高速公路的数据，通过比较随时间变化的密度和容量密度，将冲击波按类型进行了分类。下图 2.8 给出了一个例子。功能独特的冲击波类型最多有八种。
Each of these eight types are represented by the red diagonal chords that span the density curve. In each case the shockwave moves from one volume/density arrangement to another. The forming waves show an increase in density, and the recovery waves show a decrease in density. Points on the curve to the left of the maximum volume are considered to be located within an uncongested traffic state, those on the right are within a congested traffic state. The eight shockwave types are therefore defined below: 这八种类型分别用横跨密度曲线的红色对角线表示。在每种情况下，冲击波都会从一种体积/密度排列移动到另一种体积/密度排列。形成波显示密度增加，恢复波显示密度减少。曲线上位于最大体积左侧的点被认为是位于不拥堵的交通状态中，而位于右侧的点则位于拥堵的交通状态中。因此，八种冲击波类型的定义如下：
Type 1-1: Forward forming shock wave within uncongested regime 类型 1-1：在不拥堵状态下向前形成的冲击波
Type 1-2: Forward forming shock wave from uncongested regime to congested regime 类型 1-2：前向形成冲击波，从不堵车系统到堵车系统
Type 2-1: Forward recovery shock wave within uncongested regime 类型 2-1：非拥堵状态下的前向恢复冲击波
Type 2-2: Forward recovery shock wave from congested regime to uncongested regime 类型 2-2：从拥挤状态到非拥挤状态的前向恢复冲击波
Type 3-1: Backward forming shock wave within congested regime 类型 3-1：在拥塞系统内向后形成冲击波
Type 3-2: Backward forming shock wave from uncongested regime to congested regime 类型 3-2：从非拥堵状态向拥堵状态逆向形成冲击波
Type 4-1: Backward recovery shock wave within congested regime 类型 4-1：拥塞系统内的后向恢复冲击波
Type 4-2: Backward recovery shock wave from congested regime to uncongested regime 类型 4-2：从拥挤状态到非拥挤状态的后向恢复冲击波
The speed of observed shockwaves was calculated by the change in volume divided by the change in density. This is equal to the slope for each case in Figure 2.8. The type and speed of shockwaves in the minutes prior to accidents was established and statistical analysis was carried out to find a significant causal relationship. 观测到的冲击波速度是通过体积变化除以密度变化计算得出的。这等于图 2.8 中每种情况的斜率。确定了事故发生前几分钟内冲击波的类型和速度，并进行了统计分析，以找出显著的因果关系。
The study found that: 研究发现
Crashes occur more frequently when shockwaves are present 有冲击波时，碰撞发生得更频繁
the difference in the shock wave speed between the crash and non-crash cases was statistically significant 撞击和非撞击情况下的冲击波速度差异在统计学上非常显著
Lower forward shock wave speed increases the crash likelihood. 较低的前冲击波速度会增加碰撞的可能性。
More data was required to fully understand the effects of backward forming shockwaves. 要充分了解后向冲击波的影响，还需要更多的数据。
Studies of the relationship between accidents and shockwaves have been limited in scope and in detail. Nevertheless there does appear to be sufficient cause to conclude that the propagation of traffic shockwaves has a negative effect on road safety. The work of Lee Saccomano and Helinga, as well as Lee and Volpatti suggests that the speed and direction of shockwaves are the key issues relating to road safety. This paper will therefore focus its attention on these factors. 对事故与冲击波之间关系的研究在范围和细节上都很有限。不过，似乎有足够的理由得出交通冲击波的传播对道路安全有负面影响的结论。Lee Saccomano 和 Helinga 以及 Lee 和 Volpatti 的研究表明，冲击波的速度和方向是与道路安全有关的关键问题。因此，本文将重点关注这些因素。
Type 1: Forward Forming 类型 1：前进编队
Density (k) 密度 (k)
Type 3: Backward Forming 类型 3：后向成型
Density (k) 密度 (k)
Type 2: Forward Recovery 类型 2：前向恢复
Density (k) 密度 (k)
Type 4: Backward Recovery 类型 4：后向恢复
Density (k) 密度 (k)
Figure 2.8 - Shockwave Types 图 2.8 - 冲击波类型
Source: Lee and Volpatti 2010 来源：Lee and Volpatti 2010资料来源：Lee 和 Volpatti，2010 年。
2.6 Literature Review - Reflections 2.6 文献综述--思考
The literature review above presents a history of traffic flow theory split into two camps - the micro and the macro. In truth there are a number of theories which draw on both approaches simultaneously, but to detail any additional theories would achieve little. Indeed, the many discussions serve to satisfy the aims set out in Section 2.0 Literature Review : 上述文献综述介绍了交通流理论的发展历程，分为微观和宏观两大阵营。事实上，有许多理论同时借鉴了这两种方法，但详细介绍任何其他理论都将收效甚微。事实上，许多讨论都是为了满足第 2.0 节 "文献综述 "中提出的目标：
A great deal of theoretical information has been captured in the review, but the extent to which its exposure constitutes knowledge and understanding is best gauged by the reader. To be sure, the process of reviewing existing studies has impacted the path of this project. The key findings that are considered to be 本综述收集了大量理论信息，但这些信息在多大程度上构成了知识和理解，最好由读者自己来衡量。可以肯定的是，回顾现有研究的过程影响了本项目的发展方向。被认为是
Shockwaves can be generated within macroscopic or microscopic models of traffic 冲击波可在宏观或微观交通模型中产生
In all models shockwaves are understood to be density discontinuities which propagate through time and space 在所有模型中，冲击波都被理解为在时间和空间中传播的密度不连续性
There is a very wide range of approaches to traffic flow theory. Theories are becoming increasingly complex. 交通流理论的研究方法多种多样。理论变得越来越复杂。
There is no clear 'winner' - no one theory is considered to answer perfectly the questions 'how does traffic behave' and more specifically 'how do shockwaves behave' 没有明显的 "赢家"--没有一种理论能完美地回答 "交通是如何运行的"，更具体地说，是 "冲击波是如何运行的"。
There are two broad categories of shockwave - starting waves, in which the wave passes lower density through time and space, and stopping waves, in which the wave passes higher density through time and space. 冲击波有两大类--起始波和停止波，起始波是指波浪通过时间和空间时的密度较低；停止波是指波浪通过时间和空间时的密度较高。
Both types of shockwave can propagate forward and backward through space. 这两种冲击波都可以在空间中向前或向后传播。
In the scenario of an unsignalised roundabout, backward stopping shockwaves could be produced by a simple oversaturation of the approach arm, or where approach flow is lower, when there is a conflict between approach and circulating traffic. Stopping shockwaves could also move forward in the direction of the traffic if there is a particularly dense platoon dispersed upstream. Conversely if there is a relative decrease in demand then a starting wave will move forward with the direction of traffic. A starting wave will move backward upstream as queues disperse. 在没有信号灯的环岛中，进路臂的简单过饱和可能会产生向后的停车冲击波，或者在进路流量较小的情况下，当进路交通与循环交通发生冲突时，也会产生向后的停车冲击波。如果上游车流特别密集，停车冲击波也可能向车流方向前移。相反，如果需求相对减少，则起动冲击波会随着车流方向向前移动。随着队列的分散，起始波会向上游后退。
Any study which attempts to draw on these theories in its method will have to be very mathematical. Microscopic models have the benefit of providing details of shockwaves as well as individual vehicle records. 任何试图在方法上借鉴这些理论的研究都必须非常数学化。微观模型的好处是可以提供冲击波的细节以及单个车辆的记录。
The relationship between shockwaves and accident likelihood is not yet fully understood, but it is thought that shockwaves do negatively impact on road safety. Key variables associated with the impact of shockwaves are the number, speed and direction of shockwaves. 冲击波与事故发生概率之间的关系尚不完全清楚，但人们认为冲击波确实会对道路安全产生负面影响。与冲击波影响有关的关键变量是冲击波的数量、速度和方向。
Paper based traffic models do not afford users the ability to 'see' traffic operations. Instead all results are quantified numerically and appear to the layman as abstract terms. 纸质交通模型无法让用户 "看到 "交通运行情况。相反，所有结果都是用数字量化的，对于外行人来说只是抽象的术语。
Point 8 above has the effect that the literature review will be extended to include microscopic simulation software. This is to determine whether such a software could in fact be the central tool of this study as opposed to one of the many models above. 上述第 8 点的影响是，文献综述将扩展到包括微观模拟软件。这样做是为了确定，相对于上述众多模型中的一种，这种软件实际上是否可以成为本研究的核心工具。
Point 9 has directed the focus of this study to be on the number, speed and direction of shockwaves in certain conditions. Shockwave distance is also included as it relates to the number of vehicles exposed to each shockwave. 第 9 点指出，本研究的重点是特定条件下冲击波的数量、速度和方向。冲击波距离也包括在内，因为它与暴露在每个冲击波下的车辆数量有关。
3.0 Model Selection 3.0 模型选择
This paper has described the phenomenon of shockwaves at length, drawing on both macroscopic and car following conceptions of traffic flow. While it is beneficial to capture the understanding of the various approaches under consideration, clearly they cannot all be employed in pursuing the practical object of this project, as defined in the Introduction. As such a single modelling approach is to be employed by this paper. For reasons stated above, there has been too little evaluative work carried out on the main theorems exposed in the literature review of this paper, so a decision as to the optimal approach for this paper is made on more practical grounds. 本文对冲击波现象进行了详细描述，借鉴了交通流的宏观概念和车流概念。虽然对所考虑的各种方法的理解是有益的，但在实现导言中定义的本项目的实际目标时，显然不能采用所有这些方法。因此，本文将采用单一的建模方法。由于上述原因，对本文文献综述中揭示的主要定理进行的评估工作太少，因此，本文将从更实际的角度出发，决定采用哪种最佳方法。
Use of the model in this project is a means to an end, and not an end in itself. The purpose of the proposed modelling is to better understand shockwave characteristics and the conditions for shockwaves on the approach to unsignalised roundabouts at varying flow profiles. The modelling software used by this study should therefore satisfy the following criteria. 在本项目中使用该模型只是达到目的的一种手段，其本身并不是目的。拟议建模的目的是更好地了解冲击波的特性以及在不同流量剖面下接近无信号环岛时产生冲击波的条件。因此，本研究使用的建模软件应满足以下标准。
The majority of the work carried out in this study will be repetitive. The preferred model should therefore reduce manual calculation to an absolute minimum. 本研究的大部分工作都是重复性的。因此，首选模型应将手工计算减少到最低限度。
The University of Southampton must have an existing license to run the software at its Highfield Campus. 南安普顿大学必须拥有在 Highfield 校区运行该软件的许可证。
There should be expertise and familiarity with the software within the University of Southampton's Transportation Research Group (TRG), In order that this project can be fully supervised. 南安普顿大学交通研究组（TRG）应具备专业知识并熟悉该软件，以便对该项目进行全面监督。
The modelling software should have been assessed by at least one published study and have been found to represent real world traffic flow with a reasonable degree of accuracy. 建模软件应至少经过一项公开发表的研究评估，并被认定能以合理的准确度反映真实世界的交通流量。
It should be possible to alter geometric and flow inputs frequently and with ease. 应能经常轻松地更改几何和流量输入。