Lidar simulation
FORMIND allows the simulation of a Lidar scan of the forest stand at each time step (Knapp et al. 2018). The Lidar simulator generates point clouds of discrete returns as they are usually produced by small-footprint Lidar systems.
Entities, state variables and scales
The basic entities in the model are trees, Lidar pulses and Lidar returns. Trees are characterized by their position (X- and Y-coordinate), height, crown length, crown radius and crown shape (spheroid, cone or cylinder). Modeled Lidar pulses are all perfectly vertical with regular spacing between them. Thus a Lidar pulse is only characterized by its X- and Y-coordinate. Lidar returns are points in 3 dimensional space, characterized by their X-, Y- and Z-coordinate. The model works in a 3D space represented by an array of cubic voxels of a certain side length. The resolution is determined by the voxel side length, which can be chosen according to the desired spacing of Lidar pulses, as each pulse is simply represented by one vertical column of voxels.
Process overview and scheduling
In the model at first a voxel representation of the entire forest is created. This means the value of each voxel in the 3D array that falls into a tree crown or trunk of any given tree in the input list is set to one (tree voxel). All other voxels, representing free space that is not occupied by trees get the value zero, except for the forest floor (Z = 0), where each voxel gets the value one. Whether a voxel becomes a tree voxel or not depends on the tree parameters position, height, crown dimensions and shape. Precise tree positions within each 20 m x 20 m patch are assigned randomly but consistent with other FORMIND outputs, e.g. the stand visualization file. The voxel forest is hereafter scanned with a virtual Lidar. Each vertical column of voxels is considered one Lidar pulse, which can cause multiple returns. The returns are collected for each X-Y-coordinate-combination in the array.
Design concepts
The Lidar simulation follows a probabilistic approach. Instead of explicitly simulating the branches and foliage and their interaction with laser beams within the tree crowns, the model assumes that the tree crown space is a homogeneous, turbid medium. The probability to get a Lidar return from a certain point decreases with increasing distance that the laser beam has to travel through the medium before reaching that point. This is analogous to the light-extinction law after Lambert-Beer, which is also used for the light climate simulation. The only difference is that the classical Lambert-Beer equation serves to calculate remaining light intensity depending on distance, while in the discrete Lidar case the same equation is used to calculate probability $P_\text{Lid}$ for a discrete return here.
\begin{equation} P_{Lid}(d_{Lid}) = P_{0, Lid} \cdot e^{-k_{Lid} \cdot d_{Lid}} \end{equation}
Returns can only exist in tree and ground voxels. Distance $d_\text{Lid}$ that the beam has to travel through crown space before reaching a focal voxel is quantified from the count of tree voxels above the focal voxel in the same array column. $P_{0, Lid}$ represents the probability to get a return from the very upper voxel, where the laser beam hits the surface (tree or ground) for the first time. The parameter $k_{Lid}$ is the exponential extinction coefficient, which determines how fast the return probability decreases after entering the crown space. The final decision for each voxel whether it will contain a return or not is taken stochastically, based on the calculated return probability. Model outputs are tables that contain the coordinates of all tree voxels and all Lidar returns. Parameters $P_{0, Lid}$ and $k_{Lid}$ can be set independently for vegetation ($P_{0, Lid, V}$, $k_{Lid, V}$) and ground ($P_{0, Lid, G}$, $k_{Lid, G}$) voxels to adapt to the different reflectance of the forest floor. Further parameters to run the FORMIND Lidar simulator are the pulse spacing $s_{Lid}$ and the time interval between successive scans $t_{Lid}$. The Lidar simulation works with periodic boundaries, meaning that tree crowns protruding the limits of the simulated area on one side reappear on the opposite side.
Figure: Schematic visualization of the Lidar model. On the left side a forest stand of 1 ha is shown. In the center
you see a voxel representation of the stand with colors indicating Lidar
return probability from high (red) to low (blue) for each voxel. On the
right side the final simulated point cloud is shown with colors
indicating height above ground from 0 m (blue) to 40 m (orange).