HORAYZON

Package to efficiently compute terrain parameters (like horizon, sky view factor, topographic openness, slope angle/aspect) from high-resolution digital elevation model (DEM) data. The package also allows to compute shadow maps and correction factors for downwelling direct shortwave radiation for specific sun positions. Horizon computation is based on the high-performance ray-tracing library Intel© Embree. Calculations are parallelised with OpenMP (Cython code) or Threading Building Blocks (C++ code).

When you use HORAYZON, please cite:

Steger, C. R., Steger, B. and Schär, C. (2022): HORAYZON v1.2: an efficient and flexible ray-tracing algorithm to compute horizon and sky view factor, Geosci. Model Dev., 15, 6817–6840, https://doi.org/10.5194/gmd-15-6817-2022

and

Please refer to the sections Known issues and Support and collaboration in case you encounter any issues with HORAYZON.

The animation below illustrates the method applied in HORAYZON to find the terrain horizon for individual azimuth directions. Note that for performance reasons, HORAYZON determines the horizon for the first azimuth direction with a binary search (in contrast to the animation).

Package dependencies

HORAYZON depends on multiple external libraries and packages. The essential ones are listed below under Core dependencies. Further dependencies are needed to run the examples (Base dependencies for examples). The examples horizon/gridded_curved_DEM_masked.py, horizon/gridded_planar_DEM_2m.py and shadow/gridded_curved_DEM_NASADEM.py require more complex dependencies, which are listed under All dependencies for examples.

Core dependencies

• Intel Embree and Threading Building Blocks (TBB)
• Python packages: Cython, NumPy, SciPy, GeographicLib, tqdm, requests, xarray

Base dependencies for examples

• Python packages: netCDF4, Matplotlib, Pillow, Skyfield, pyproj, IPython

All dependencies for examples (masking and high-resolution DEM examples; GDAL dependency)

• Python packages: Shapely, fiona, PyGEOS, scikit-image, Rasterio, Trimesh
• heightmap meshing utility (hmm)

Installation

HORAYZON has been tested with Python 3.10 (Linux) and Python 3.11 (Mac OS X). It is recommended to install dependencies via Conda, which covers all dependencies except hmm. Alternatively, HORAYZON can also be installed without Conda (by e.g. using pip to install Python packages). Installation via Conda can be accomplished as follows for different platforms:

Linux

Installation requires the GNU Compiler Collection (GCC). Create an appropriate Conda environment

Core dependencies

conda create -n horayzon_core -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray

Base dependencies for examples

conda create -n horayzon_base -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray netcdf4 matplotlib pillow skyfield pyproj ipython

All dependencies for examples (masking and high-resolution DEM examples; GDAL dependency)

conda create -n horayzon_all -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray netcdf4 matplotlib pillow skyfield pyproj ipython shapely fiona pygeos scikit-image rasterio trimesh

and activate this environment. The HORAYZON package can then be installed with:

git clone https://github.com/ChristianSteger/HORAYZON.git
cd HORAYZON  
python -m pip install .

Mac OS X

HORAYZON is compiled with Clang under Mac OS X. As the Apple-provided Clang does not support OpenMP, an alternative Clang with OpenMP support has to be installed. This can be done via Conda. Create an appropriate Conda environment

Core dependencies

conda create -n horayzon_core -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray c-compiler openmp

Base dependencies for examples

conda create -n horayzon_base -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray netcdf4 matplotlib pillow skyfield pyproj ipython c-compiler openmp

All dependencies for examples (masking and high-resolution DEM examples; GDAL dependency)

conda create -n horayzon_all -c conda-forge embree3 tbb-devel cython numpy scipy geographiclib tqdm requests xarray netcdf4 matplotlib pillow skyfield pyproj ipython shapely fiona pygeos scikit-image rasterio trimesh c-compiler openmp

and activate this environment. The HORAYZON package can then be installed with:

git clone https://github.com/ChristianSteger/HORAYZON.git
cd HORAYZON
python -m pip install .

Optional installation of hmm

hmm depends on glm, which can also be installed via Conda

conda install -c conda-forge glm

Alternatively, glm can also be built manually from source. hmm can then be downloaded with

git clone https://github.com/fogleman/hmm.git
cd hmm

The following two lines in hmm's Makefile might have to be adapted to (the include directory in the first line is valid in case glm was installed with Conda):

COMPILE_FLAGS = -std=c++11 -flto -O3 -Wall -Wextra -Wno-sign-compare -march=native -lGL -lglut -lGLEW -I<path to directory 'include' of conda environment>
INSTALL_PREFIX = <binary install path>

Finally, hmm can be installed with

make
make install

Installation without Conda

HORAYZON can also be built without Conda but this requires some additional manual steps. If not already available, the following two external libraries Intel Embree and Threading Building Blocks (TBB) have to be installed. This can be done either via a package manager (APT, MacPorts, etc.) or by manually building them from source. Afterwards, the required Python packages have to be installed (for instance with pip) and the HORAYZON package can be downloaded:

git clone https://github.com/ChristianSteger/HORAYZON.git
cd HORAYZON

The setup file setup_manual.py must then be adapted to specify the include and library paths for the external libraries and to select a compiler to build HORAYZON. Finally, the HORAYZON package can be installed with:

mv setup_manual.py setup.py
python -m pip install .

Usage

The usage of the packages is best illustrated by means of examples, which can either be run in a Python IDE (like PyCharm or Spyder) or in the terminal. To run the examples, the path path_out must be adapted in the example script to a location that provides enough disk space. For the example horizon/gridded_planar_DEM_2m.py, the path to the hmm executable (hmm_ex) has to be additionally adapted. All input DEM or auxiliary data required for running the examples is downloaded automatically. When HORAYZON tries to download auxiliary data for the first time, a local path for the data has to be provided by the user. This path is saved in the text file path_aux_data.txt, which is stored in the directory to which the HORAYZON package was installed. In case this path is unknown, it can be found by running

import horayzon
print(horayzon.__file__)

in Python. If the auxiliary data is later on moved manually to a new directory, the path in path_aux_data.txt has to be adapted accordingly.

Examples: Terrain parameters (slope, horizon and sky view factor)

Two terrain horizon functions are available, horizon_gridded() and horizon_locations(). The former function allows to compute horizon for gridded input while the latter allows to compute horizon for arbitrary selected locations. The second function can optionally output the distance to the horizon. The following five examples are provided:

• horizon/gridded_curved_DEM.py: Compute topographic parameters from SRTM (geodetic coordinates, ~90 m resolution) for a ~50x50 km example region in the European Alps. Earth's surface curvature is considered. Plot output of this script is shown below.
• horizon/gridded_planar_DEM.py: Compute topographic parameters from swisstopo DHM25 (map projection, 25 m resolution) for a ~25x40 km example region in Switzerland. Earth's surface curvature is neglected.
• horizon/locations_curved_DEM.py: Compute topographic parameters (additionally distance to horizon) from SRTM (geodetic coordinates, ~90 m resolution) for 11 locations in Switzerland. Earth's surface curvature is considered. Plot output of this script for one location is shown below.
• horizon/gridded_curved_DEM_masked.py: Compute topographic parameters from SRTM (geodetic coordinates, ~90 m resolution) for South Georgia in the South Atlantic Ocean. Earth's surface curvature is considered. DEM grid cells, which are at least 20 km apart from land, are masked to speed-up horizon computation.
• horizon/gridded_planar_DEM_2m.py: Compute gridded topographic parameters from swissALTI3D (map projection, 2 m resolution) for a 3x3 km example region in Switzerland. Earth's surface curvature is neglected. The outer DEM domain is simplified and represented by a triangulated irregular network (TIN) to reduce the large memory footprint of the DEM data.

A remark on sky view factor and related parameters
The term sky view factor (SVF) is defined ambiguously in literature. In Zakšek et al. (2011), it refers to the solid angle of the (celestial) hemisphere. We call this parameter visible sky fraction and its computation is performed with the function topo_param.visible_sky_fraction(). In applications related to radiation, the SVF is typically defined as the fraction of sky radiation received at a certain location in case of isotropic sky radiation (see e.g. Helbig et al., 2009). This parameter is called sky view factor in our application and its computation is performed with the function topo_param.sky_view_factor(). Additionally, the positive topographic openness (Yokoyama et al., 2002) can be computed with the function topo_param.topographic_openness().

Examples: Shadow map and shortwave correction factor

The module shadow allows to compute shadow maps and correction factors for downwelling direct shortwave radiation for arbitrary terrains and sun positions. This module was not part of the initial HORAYZON release and is thus not described in the reference publication. A more detailed description is therefore provided here. To compute gridded shadow maps or shortwave correction factors, a class shadow.Terrain must first be created and initialised. In this step, the gridded terrain input is first converted to a triangle mesh and these triangles are then stored in a bounding volume hierarchy (BVH) to perform ray casting efficiently. During initialisation, and optional mask can be provided to ignore certain grid cells and a flag to consider atmospheric refraction can be enabled. The two methods Terrain.shadow() and Terrain.sw_dir_cor() can then be called for arbitrary sun positions. The output of the method Terrain.shadow() is encoded as follows: 0: illuminated, 1: self-shaded, 2: terrain-shaded, 3: not considered (respectively masked). The correction factors for downwelling direct shortwave radiation is computed with the method Terrain.sw_dir_cor() according to Müller and Scherer (2005). This factor can be applied to radiation output from a regional climate or general circulation model, in which radiation is only simulated along the vertical dimension and all grid cells are assumed to have a horizontal surface. The correction factor accounts for all terrain-induced modifications in radiation, like self/terrain-shading, changes in angles between the sun and the surface's normal vector and the geometric surface enlargement of grid cells due to sloping surfaces. According to Equation (2) in Müller and Scherer (2005), the correction factor is computed as

$$f_{cor} = \left( \dfrac{1.0}{\vec{h} \cdot \vec{s}} \right) \left( \dfrac{1.0}{\vec{h} \cdot \vec{t}} \right) \ {mask}_{shadow} \ \left( \vec{t} \cdot \vec{s} \right)$$

where vector h is the normal of the horizontal surface, vector t the normal of the tilted surface, vector s the sun position vector and mask_shadow the terrain-shading mask (0: shadow, 1: illuminated). All above vectors represent unit vectors. The same equation for the correction of downwelling direct shortwave radiation is applied in Manners et al. (2012).

• shadow/gridded_curved_DEM_SRTM.py: Compute shadow map and shortwave correction factor from SRTM (geodetic coordinates, ~90 m resolution) for South Georgia in the South Atlantic Ocean for a day in southern-hemisphere winter. Earth's surface curvature and atmospheric refraction are considered. Plot output of this script is shown below.
• shadow/gridded_curved_DEM_REMA.py: Compute shortwave correction factor from REMA (map projection, ~100 m resolution) for an example region in Antarctica for a day in southern-hemisphere summer. Earth's surface curvature and atmospheric refraction are considered and ocean grid cells are ignored (masked).
• shadow/gridded_planar_DEM_artificial.py: Compute shortwave correction factor from artificial topography (hemispherical mountain in the centre). The illumination source (sun) rotates once around the centre.
• shadow/gridded_curved_DEM_NASADEM.py Compute shortwave correction factor from NASADEM (geodetic coordinates, ~30 m resolution) for an example region in the Karakoram for a day in northern-hemisphere winter. Earth's surface curvature is considered and atmospheric refraction ignored. All non-glacier grid cells are masked to speed-up computation. An NASA Earthdata account is required and wget has to be set to download NASADEM data.

Atmospheric refraction
Close to the unobstructed terrestrial horizon, the position of the sun is significantly influenced by atmospheric refraction. The solar elevation angle of the true position is thereby lower than the apparent position. We included an option (refrac_cor=True) to account for this effect by applying the formula of Sæmundsson (1986). This formula is also presented in Meeus (1998) and reads

$$r = \frac{1}{60} \left(1.02 \cdot \cot \left(h_{t} + \frac{10.3}{h_{t} + 5.11}\right) + 0.0019279 \right) \cdot \left(\frac{p}{101} \frac{283}{273 + T}\right)$$

with r representing atmospheric refraction (degree), h_t the sun's true elevation angle (degree), p atmospheric pressure (kPa) and T temperature (° C). Note that the function argument of cot must be provided in radian. Atmospheric refraction increases with increasing air pressure and decreasing temperature and is only significant for very low solar elevation angles, as illustrated in the below figure. The dotted lines represent the raw output according to the above equation. We keep refraction correction constant for elevation angles smaller than -1.0°. At sea level, we assume a temperature of T₀ = 10° C and an atmospheric pressure of p₀ = 101.0 kPa. These quantities are extrapolated to higher elevations with a constant linear temperature lapse rate and the hydrostatic assumption according to the following two equations

$$T(z) = T_{0} - L \cdot z$$

$$p(z) = p_{0} \cdot \left(\frac{T_{0} - L \cdot z}{T_{0}}\right)^{\frac{g}{R_{d} \cdot L}}$$

with g representing the acceleration due to gravity (9.81 m ^-2), R_d the gas constant for dry air (287.0 J K^-1 kg^-1) and L the lapse rate (0.0065° C m^-1). These assumptions yield a temperature of -9.5° C and an atmospheric pressure of ~70 kPa for an elevation of 3000 m a.s.l. According to the above figure, changes in atmospheric pressure dominate the influence on atmospheric refraction, which results in less significant refraction effects for elevated areas like mountains.

Digital elevation model and auxiliary data

Digital elevation model (DEM) data is available from various sources, e.g.:

• SRTM
• DHM25
• swissALTI3D
• NASADEM
• MERIT
• USGS 1/3rd arc-second DEM

Auxiliary data, like geoid undulation data (EGM96 and GEOID12A), coastline polygons (GSHHG) or glacier outlines (GAMDAM) are available from here:

• EGM96
• GEOID12A
• GSHHG
• GAMDAM

Known issues

The below list contains known issues with HORAYZON, which will be addressed in a future release:

• Some inconsistencies in user-defined input arguments are currently not checked due to performance reasons. A known problematic argument pair is dist_search and elev_ang_low_lim in the functions horizon_gridded() and horizon_locations(). Horizon elevation angles can be distinctively negative for very small horizon search distances (e.g. 1 km) and elevated positions like mountain peaks. Such low elevation angles fall below the default setting for elev_ang_low_lim of -15.0°. To prevent the algorithm from being stuck in a infinite loop, a smaller value for elev_ang_low_lim has to be chosen (e.g. -89.0°).

Comparison with other algorithm

Another high-performance and parallelised algorithm to compute terrain horizon is presented in Dozier (2022). A brief comparison between this algorithm and HORAYZON can be found here.

References

• Dozier, J. (2022): Revisiting the topographic horizon problem in the era of big data and parallel computing, IEEE Geosci. Remote Sens. Lett., 19, 1-5, https://doi.org/10.1109/LGRS.2021.3125278
• Helbig, N., Löwe, H. and Lehning, M. (2009): Radiosity Approach for the Shortwave Surface Radiation Balance in Complex Terrain, J. Atmos. Sci., 66(9), 2900-2912, https://doi.org/10.1175/2009JAS2940.1
• Manners, J., Vosper, S.B. and Roberts, N. (2012), Radiative transfer over resolved topographic features for high-resolution weather prediction. Q.J.R. Meteorol. Soc., 138: 720-733. https://doi.org/10.1002/qj.956
• Meeus (1998). Astronomical algorithms (Second edition). Richmond, Va.: Willmann-Bell. pp. 105–108. ISBN 0943396611.
• Müller, M. D. and Scherer, D. (2005): A Grid- and Subgrid-Scale Radiation Parameterization of Topographic Effects for Mesoscale Weather Forecast Models, Mon. Weather Rev., 133(6), 1431-1442, https://journals.ametsoc.org/view/journals/mwre/133/6/mwr2927.1.xml
• Sæmundsson (1986). Astronomical Refraction. Sky and Telescope. 72: 70.
• Yokoyama, R., Shirasawa, M. and Pike, R. J. (2002): Visualizing Topography by Openness: A New Application of Image Processing to Digital Elevation Models, Photogramm. Eng. Remote Sens., 68, 257-265.
• Zakšek, K., Oštir, K. and Kokalj, Ž. (2011): Sky-View Factor as a Relief Visualization Technique, Remote Sens., 3(2):398-415, https://doi.org/10.3390/rs3020398

Support and collaboration

In case of issues or questions, contact Christian R. Steger (christian.steger@env.ethz.ch). Please report any bugs you find in HORAYZON. You are welcome to fork this repository to modify the source code - we are open to consider pull requests for future HORAYZON versions/releases.