pairs $(\rho_j,Q_j),\;j=1,\dots n$ are given from long-term measurements (commonly obtained via stationary sensors). As visible in the right panel of Fig. \[fig:fd\_lwr\_arz\], these data (gray dots) tend to exhibit a relatively clear functional relationship between $\rho$ and $Q$ for low densities. In turn, for medium densities, a significant spread is visible, i.e., a single $\rho$-value corresponds to many different flow rates $Q$. Finally, for large densities, very few data points are available at all.

Data-Fitting for the LWR and ARZ Models
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The first-order LWR model must represent these data via a single function $Q(\rho)$. As the spread of the data cannot be captured, it is reasonable to find a function that lies “in the middle” of the cloud of data points. Specifically, we employ the approach presented in [@FanSeibold2013]. First, since the stagnation density $\rho_\text{max}$ is not represented well via data, we prescribe it as a fixed constant, given by a typical vehicle length of 5 meters, plus 50% of additional safety distance, $$\rho_\text{max} = \frac{\text{number of lanes}}
{\text{typical vehicle length}\times\text{safety distance factor}}
= \frac{\#\text{lanes}}{7.5\text{m}}\;.$$

Second, a three-parameter family of smooth and strictly concave flow rate curves is selected as $$\label{eq:flow_rate_curve}
Q_{\alpha,\lambda,p}(\rho) = \alpha{\left(a+(b-a)\rho/\rho_\text{max}-\sqrt{1+y^2}\right)}\;,$$ where $$\begin{aligned}
a = \sqrt{1+\left(\lambda 