Different travel time measurements and the space of potentially realized trajectories that result in each travel time

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Context 1

... the idea, presented in this extension, is to introduce a dynamic weight that is assigned to gridded BT speed data, which express the trustworthiness of the computed grid speeds. The trustworthiness is influenced by the detector spacing and the measured travel time: Assume a vehicle needs time ∆t to travel distance ∆x (see Fig. 3). It has a maximum speed of v max and a minimum speed of v min . Further assume that the vehicle is not standing, such that v min > 0. Then, for an observer who only measured ∆t and ∆x, it is not known where the vehicle was positioned, and at what speed it was driving, while passing the measured distance in measured time. From the ...

Context 2

... and ∆x, it is not known where the vehicle was positioned, and at what speed it was driving, while passing the measured distance in measured time. From the observer's perspective, however, given the assumed minimum and maximum speed, the vehicle's position can be restricted to a certain space-time area. This area is depicted as a parallelogram in Fig. 3, along with three examples of potential vehicle trajectories. Each potential trajectory can be described as a function of the vehicle's position x(t), and its corresponding velocity v c (t). As illustrated, a medium travel time allows for strong deviations of v c (t) over time, whereas low travel times restrict v c (t) to higher ...