This method calculates the mean value of flow and density over time. A reference line is taken and studied over a fixed period of time $$\Delta {t}$$.
Using this method we can obtain the pedestrian flow $$J$$ and the velocity $$v_i$$ of each pedestrian passing the reference line directly. Thus, the flow over time $$\langle J \rangle_{\Delta t}$$ and the time mean velocity $$\langle v \rangle_{\Delta t}$$ can be calculated as

$\langle J \rangle_{\Delta t}=\frac{N^{\Delta t}}{t_N^{\Delta t} - t_1^{\Delta t}}\qquad \text{and} \qquad \langle v \rangle_{\Delta t}=\frac{1}{N^{\Delta t}}\sum_{i=1}^{N^{\Delta t}} v_i(t),$

where $$N^{\Delta t}$$ is the number of persons passing the reference line during the time interval $$\Delta {t}$$.

$$t_N^{\Delta {t}}$$ and $$t_1^{\Delta {t}}$$ are the times when the first and last pedestrians pass the location in $$\Delta {t}$$.

The time mean velocity $$\langle v \rangle_{\Delta t}$$ is defined as the mean value of the instantaneous velocities $$N^{\Delta t}$$ pedestrians.

$$v_i(t)$$ is calculated by use of the displacement of pedestrian $$i$$ in a small time interval $$\Delta t^\prime$$ around $$t$$:

$v_i(t)=\frac{\vec{x_i}(t+\Delta t^\prime/2)-\vec{x_i}(t-\Delta t^\prime/2))}{\Delta t^\prime}.$
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