| | 15 | == Representation of radiation in RTM == |
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| | 17 | This is only a brief overview of the most important RTM concepts. A full description of the RTM model is available in [#krc2021 Krč et al., 2021]. |
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| | 19 | The discretization of RTM uses the same Cartesian grid as the rest of the PALM model. Each radiative quantity is modelled as a singular value per surface discretization unit (face), and the propagation of radiation is described as interactions between mutually visible faces. |
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| | 21 | The model considers all reflections and emissions to be Lambertian (i.e. ideally diffuse), following Lambert's cosine law whereby the amount of radiation leaving the surface in one direction is proportional to the cosine of the angle θ between that direction and the surface normal. The interaction between faces can therefore be described similarly for reflection and for thermal emission. |
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| | 23 | For any two mutually visible faces ''i'' and ''j'', the view factor (VF) ''F,,i→j,,'' is the fraction between the radiant flux originating from face ''i'' that strikes face ''j'' and the total radiant flux leaving face ''i''. The view factor values carry all the information about the geometry of the urban layer necessary for calculating propagation of reflected and emitted light among surfaces. Once they are known, calculation of the instantaneous fluxes can be reduced to simple vector multiplication. Determining the view factor values consists of multiple steps. |
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| | 25 | 1. Establishing mutual orientation and position. In the rectangular grid, this is a matter of performing multiple coordinate comparisons to find out whether, for each face, the other face lies in the half-space above the plane of the first face, i.e. whether its angle θ is less than π/2. |
| | 26 | |
| | 27 | 2. Determining obstacles on the ray path between the faces. The obstacles may be fully opaque (terrain, buildings) or partially transparent, in which case a fraction of the radiant flux between the faces is absorbed. In RTM, the only partially transparent obstacle is the grid-resolved plant canopy, which is represented as a 3-D field of leaf area density (LAD). The fraction of the radiant flux allowed to pass through the obstacle and the radiant flux carried by the ray upon striking the obstacle is called transmittance. For the plant canopy, it depends on the length of the ray's intersection with the respective PCGB, the LAD value at that PCGB and the extinction coefficient. |
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| | 29 | 3. Calculating the actual view factor value. |
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| | 31 | The second step is implemented in RTM using a ray-tracing algorithm. This process is computationally complex, as it performs calculations involving each grid box that each traced ray intersects, and it can also cause very high demands on the interprocess communication. In PALM, each parallel process is responsible for modelling a horizontally divided subdomain within the modelled domain, and most of the data stored locally are limited to the extent of the subdomain. The access to the values in other subdomains carried by MPI interprocess communication is significantly slower than similar local memory access. Depending on the domain size and geometry, each traced ray may cross many subdomains. Due to this complexity, the ray-tracing task takes place during the model initialization phase before the actual simulation of time steps begins. The values representing the view factors and other relevant data are precomputed, exchanged among the parallel processes and stored in such a way that the number of calculations and MPI communications performed during computation of time steps is minimized. |
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| | 33 | The view from each face is discretized using, by default, the ''angular discretization scheme'', which divides the view into a fixed number of directions specified by uniformly distributed azimuth and elevation angles. Ray tracing is performed towards this fixed set of directions with considerable optimization due to the fact that multiple rays of this set share an identical horizontal direction. For each ray, the face that covers the first detected obstacle (terrain or building) is used to create a view factor entry. Its view factor value represents exactly the portion of the view corresponding to its direction segment (the section of azimuths and elevations instead of being determined by the other face's size and position). The following figures depict the geometry of the discretization for a horizontal and a vertical face. |
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| | 35 | [[Image(ray_h.png,625px)]] [[Image(ray_v.png,450px)]] |
