First order velocity based travel time model

A general first-order model relating directional derivatives of travel time with speed at each point in the time-space domain is used in speed based travel time estimation. Due to its partial differential equation (PDE) form, this model makes it possible to estimate travel times without the need for trajectory construction based on speeds which is a cumbersome integration operation. A forward time backward space (FTBS) finite difference scheme is presented to approximate the general model. While the general model and its corresponding FTBS solution scheme are designed for smooth situations, the traffic stream is replete with discontinuities especially as congestion increases. For these situations an equivalent conservative model based on the first order continuum traffic flow model with Greenshields flux is proposed. A Godunov scheme is proposed to approximate the conservative model. US-101 dataset from NGSIM project is used to illustrate the performance of the proposed models. Three factors impacting the performance of the proposed schemes that is underlying speed estimate accuracy, traffic congestion level and discretization level are considered.