  In this paper, we describe a relation between a microscopic traffic cellular
automaton (TCA) model (i.e., the stochastic TCA model of Nagel and
Schreckenberg) and the macroscopic first-order hydrodynamic model of Lighthill,
Whitham, and Richards (LWR). The innovative aspect of our approach, is that we
explicitly derive the LWR's fundamental diagram directly from the STCA's rule
set, by assuming a stationarity condition that converts the STCA's rules into a
set of linear inequalities. In turn, these constraints define the shape of the
fundamental diagram, which is then specified to the LWR model. Application of
our methodology to a simulation case study, allows us to compare the
tempo-spatial behavior of both models. Our results indicate that, in the
presence of noise, the capacity flows in the derived fundamental diagram are
overestimations of those of the STCA model. Directly specifying the STCA's
capacity flows to the LWR fundamental diagram, effectively remedies most of the
mismatches between both approaches. Our methodology sees the STCA complementary
to the LWR model and vice versa, so the results can be of great assistance when
interpreting the traffic dynamics in both models. Especially appealing, is the
fact that the STCA can visualize the higher-order characteristics of traffic
stream dynamics, e.g., the fans of rarefaction waves.
