On sequential data assimilation for scalar macroscopic traffic flow models

We consider the problem of sequential data assimilation for transportation networks using optimal filtering with a scalar macroscopic traffic flow model. Properties of the distribution of the uncertainty on the true state related to the specific nonlinearity and non-differentiability inherent to macroscopic traffic flow models are investigated, derived analytically and analyzed. We show that nonlinear dynamics, by creating discontinuities in the traffic state, affect the performances of classical filters and in particular that the distribution of the uncertainty on the traffic state at shock waves is a mixture distribution. The non-differentiability of traffic dynamics around stationary shock waves is also proved and the resulting optimality loss of the estimates is quantified numerically. The properties of the estimates are explicitly studied for the Godunov scheme (and thus the Cell-Transmission Model), leading to specific conclusions about their use in the context of filtering, which is a significant contribution of this article. Analytical proofs and numerical tests are introduced to support the results presented. A Java implementation of the classical filters used in this work is available on-line at http://traffic.berkeley.edu for facilitating further efforts on this topic and fostering reproducible research.