## 2 Literature Reviews  
The characterization and estimation of the road traffic state play an important role in relieving traffic congestion or optimizing the use of the road infrastructure. The traffic  
Figure 1: An illustration of the study system. Figure 2: MFD from experimental data for departure vehicles at the airport: (**a**) V-K relationship; (**b**) V-Q relationship; (**c**) Q-K relationship; (**d**) Q-K-V relationship. impedance (the relationship between vehicle travel time and traffic flow) and the fundamental diagram are theories that characterize the state of traffic. Recently, some new results have emerged from research on the characterization and estimation of the traffic state. For example, [PERSON] et al. [3] proposed an equation to characterize the relationship between vehicle travel time and traffic flow belonging to the sigmoid family on most of the road sections in the city of Valencia, Spain, which does not fit the general theory of vehicle flow, namely, by applying logistic regression, and the results were able to significantly improve the curve fitting of these road sections. [PERSON] et al. [4] constructed a real-time traffic state estimation framework for urban roads based on an artificial neural network by acquiring road traffic data with the help of cellular networks. The validation using simulation and real data shows that the model has stable and reliable performance in predicting the real-time traffic condition of urban roads. The discovery of the basic parameter relationships of traffic flow opened the study of traffic flow theory. The basic diagram describes the interchange relationship of traffic flow, density and speed, and reflects the change in traffic state. Since [PERSON] pioneered the study of traffic flow theory, many scholars have studied the shape of the fundamental diagram and proposed a variety of traffic flow models, as shown in Table 1.  
[PERSON] [5] earlier proposed a linear speed-density relationship model, from which a parabolic-shaped flow-density model can be derived. [PERSON] [6] derived a logarithmic speed-density model based on fluid dynamic principles and found that this model is more consistent with the data under traffic congestion. [PERSON] [7] obtained the exponential speed-density model by studying the measured traffic data, and further study found that it is not applicable to the traffic condition on fast roads, but to less dense traffic situations. The GHR_M1 model is a bell-shaped speed-density model proposed by [PERSON] et al. [8] The model's estimate of the speed of the free flow is usually lower than the actual value. The GHR_M2 model is a family of model curves proposed by [PERSON] and [PERSON] [9] with parameters \(n\) greater than 0 and with different shapes for three cases: \(n<1\), \(n=1\), \(n>1\). The Draw model proposed by [PERSON] [10] is similar to the GHR_M2 model in that it introduces an additional parameter that allows for greater degrees of freedom in data fitting. The Kuhne and Rodiger model is a model proposed by [PERSON] and [PERSON] [11] to simulate highway traffic under heavy traffic conditions. The model takes into account the relaxation of the static speed and density relationship and the prediction of downstream traffic conditions. The S3 model is an S-shaped speed-density model containing three parameters proposed by [PERSON] et al. [12] The model is able to describe the relationship between the variation in traffic speed and density on highways under a wide range of possible density conditions. The two-fluid model is a model proposed by [PERSON] and [PERSON] [13] based on the kinetic theory of multilane highway traffic for studying the transition of traffic to a collective flow regime under high-density conditions. The car-following model is an application of differential equations to describe the vehicle-following motion law, and the NF model is derived from [PERSON]'s car-following model [14].