## 3 Method
### Wind-Field Propagation
This approach assumes that the measured air parcels evolve downstream along with the local streamwise velocity. This implies that the air parcels do not adhere to [PERSON]'s frozen turbulence hypothesis that assumes advection with a global mean velocity [PERSON] et al. (2015). Like the approach of [PERSON] et al. (2014), the following approach can be described using a semi-Lagrangian scheme. At each time step (\(t\)), the velocity data on a grid point is interpreted as a parcel of air that is free to move with its own velocity (\(u\in\mathbb{R}\)) and direction (\(\vec{u}\in S^{1}\subset\mathbb{R}^{2}\)), with \(S^{1}\Big{\{}(\chi,\gamma)\in\mathbb{R}^{2}\ \Big{|}\ \sqrt{\chi^{2}+\gamma^{2}}=1 \Big{\}}\). This can be understood as an interpretation of [PERSON] (1955) that applies to small spatial and temporal scales. The advection of each parcel can be approximated in a discretised fashion using the following equations:  
\[\frac{\partial u(\chi,t)}{\partial t}=-u(\chi,t)\cdot\
abla u(\chi,t), \tag{15}\]  
\[u(\chi,t+\Delta t)=u(\chi,t)-\Delta t\cdot u(\chi,t)\cdot\frac{\delta u(\chi,t)}{\delta\chi}, \tag{16}\]\[u(\chi,t+\Delta t)=u(\chi,t)-\Delta t\!\cdot\!u(\chi,t)\!\cdot\!\frac{u(\chi,t)-u( \chi-\Delta\chi,t)}{\Delta\chi}, \tag{17}\]  
with \(\frac{\Delta\chi}{\Delta t}=u(\chi,t)\) respectively \(\Delta\chi=u(\chi,t)\!\cdot\!\Delta t\). Note that this equation is given in only one dimension because it is projected on the affine subspace, \(\chi+\lambda\overset{\rightarrow}{u}\subset\mathbb{R}^{2}\), with \(\lambda\in\mathbb{R}\). This representation gives:  
\[u(\chi,t+\Delta t)=u(\chi,t)-(u(\chi,t)-u(\chi-\Delta t\!\cdot\!u(\chi,t),t)), \tag{18}\]  
\[u(\chi,t+\Delta t)=u(\chi-\Delta t\!\cdot\!u(\chi,t),t), \tag{19}\]  
where \(\Delta\chi\) is the distance that the given air packet travels in one time step (\(\Delta t\)). If we define \(\chi_{old}\chi-\Delta t\!\cdot\!u\overset{\rightarrow}{u}\) as the initial position of the packet on the regular Cartesian grid, Equation (19) shows that, at the next time step, \(t_{new}t+\Delta t\), each parcel transports its own velocity and direction to its new location via advection (\(\chi=\chi_{old}+\Delta t\!\cdot\!u\overset{\rightarrow}{u}\)). This means that parcels on the regular Cartesian grid are displaced to an irregular grid. To resolve the new velocity field in the original grid, the velocities in the intermediate grid are interpolated onto the initial grid using natural-neighbour interpolation [40]. To give an example of this method, we initialise this space-time conversion approach using a scan from Section 2.2 at \(t_{n}\) and iterate it with selected temporal resolutions (\(\Delta t\)) to the time interval of the subsequent scan (\(t_{n+1}\)). Comparing the iterated and interpolated scans at \(t_{n+1}\) with the processed LiDAR scan at \(t_{n+1}\), the flow structures are subjected to numerical diffusion and smoothed out by the subsequent interpolation. To minimise the multiplication of errors with multiple iterations, we applied a mixed propagation approach with the temporal weighted average of forward- and backward-oriented propagations. In the backwards-oriented propagation, we initialise the process with a negative time step (\(-\Delta t\)) that causes the air parcel at \(t_{n+1}\) to propagate backwards in time to \(t_{n}\). The sinusoidal function, \(w_{n}(t)\), is used to calculate the weighted average of forward and backward propagations in order to minimise numerical diffusion and to guarantee continuous advection with no unphysical gaps. We are aware that fluid-mechanical mixing processes are irreversible. We assume, however, that the spatial scale of the mixing process is considerably smaller than the spatial resolution of the measurements, so irreversibility is not a concern. We define \(t\in\{\mathbb{R}\mid t_{n}\leq t\leq t_{n+1}\}\) and:  
\[w_{n}(t)=\left\{