In the past, vehicle arrivals on the road links were always detected using loop coil detectors. This type of traffic sensor is buried under the ground, which causes great damage to the road surface and requires a large amount of maintenance. The geomagnetic detector is not only stable and reliable with high detection accuracy, but also convenient to install and maintain. These two magnetic detectors can obtain similar traffic flow information, such as traffic flow rate and occupancy rate. As an alternative to the loop coils, the geomagnetic detectors are developing into an important fixed urban road detector.

Geomagnetic detector data (GDD) belong to point traffic flow data, and the certain mathematical models need to be constructed to obtain the corresponding average travel time. The previous studies [

26,

27] show that the link travel time series in highway and urban road environments are greatly affected by the fluctuation of traffic flows, but they exhibit different function relationships. The BPR (Bureau of Public Roads) function [

28] is a travel time estimation model with traffic flow rate as an independent variable for highway environment. Its mathematical formula is as follows:

where

${T}_{ij,GDD}^{t}$ is the average travel time estimate of observed link

$({v}_{i},{v}_{j})$ using the GDD at the timestep

t;

${T}_{ij,f}$ is the travel time of observed link (

v_{i},

v_{j}) in free flow state;

${q}_{ij,GDD}^{t}$ is the actual traffic flow of observed link

$({v}_{i},{v}_{j})$ obtained from the GDD at the timestep

t;

c_{ij} is the capacity of observed link

$({v}_{i},{v}_{j})$;

$\alpha $ and

$\beta $ are impedance parameters. The BPR function shows three important relationship characteristics: (i) the link travel time is close to the free-flow travel time when actual traffic flow is small enough; (ii) the link travel time varies slowly and is proportional to traffic flow when actual flow is far less than the link capacity; (iii) the link travel time increases rapidly with the change of traffic flow when actual flow approaches or exceeds capacity. Unlike the highway environment, there are signal controls in urban road networks. As traffic congestion is increasingly heavier, urban link travel time will not get continuous growth. This means that when traffic flow exceeds the capacity and road link reaches the certain congested level, the flow begins to decrease and the travel time increases to a stable high value. So, the BPR function model cannot be directly applied to urban roads, and the uniformly calibrated BPR model achieves poor estimation in the congested state. In view of this, the BPR model is calibrated by differentiating traffic conditions, so as to make better use of the GDD to estimate urban link travel time [

29]. This paper considers the product of traffic flow and occupancy rate from the GDD as road traffic state index. The specific calculation formula is as follows:

where

${I}_{ij,GDD}^{t}$ is traffic state index of observed link

$({v}_{i},{v}_{j})$ using the GDD at the timestep

t;

${q}_{ij,GDD}^{t}$ and

${o}_{ij,GDD}^{t}$ are actual flow and occupancy rate of observed link

$({v}_{i},{v}_{j})$ from the GDD at the timestep

t. Taking the historical GDD and LPR data of Jingshi Road in Jinan as an example, the results show that the traffic state index is relatively consistent with the trend of link travel time series, as shown in

Figure 2. The link traffic state is classified into three categories according to the change trend of this index: (i) when

$0\le {I}_{ij,GDD}^{t}<40$, the traffic state index is low and stable, and the observed link is smooth at the timestep

t; (ii) when

$40\le {I}_{ij,GDD}^{t}<200$, the traffic state index increases and the observed link is in a blocked state at the timestep

t; (iii) when

${I}_{ij,GDD}^{t}\ge 200$, the traffic state index is high and the observed link is in a congested state at the timestep

t.

The BPR function is calibrated in the above three states of smooth, blocked and congested, respectively. The calibrated BPR function is as follows: