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Sky View Factor (SVF)

Fraction of the sky hemisphere visible from each point. Determines how much diffuse sky radiation and longwave sky emission reaches the surface.

Reference: Lindberg et al. (2008) Section 2.1, Lindberg & Grimmond (2011)

Equation

SVF is the ratio of radiation received from the sky to that from an unobstructed hemisphere:

SVF = Ω_sky / 2π

Where Ω_sky is the solid angle of visible sky (steradians).

Patch-Based Calculation Method

Reference: Robinson & Stone (1990) "Solar Radiation Modelling in the Urban Context", Building and Environment 25(3):201-209.

SOLWEIG uses the patch-based method where the sky hemisphere is divided into discrete angular patches (annuli). SVF is computed by testing visibility to each patch and weighting by the patch's solid angle.

Patch Configuration

The sky is divided into concentric annuli (altitude bands) from 0° to 90° elevation. Each annulus is further subdivided into azimuthal patches. Standard configurations:

  • Option 1 (145 patches): Coarse for fast computation
  • Option 2 (153 patches - default): Balance of accuracy and speed
  • Option 3 (305 patches): Fine for high accuracy
  • Option 4 (609 patches): Research-grade resolution

For option 2 (153 patches):

  • 8 altitude bands: 6°, 18°, 30°, 42°, 54°, 66°, 78°, 90°
  • Azimuthal divisions per band: 31, 30, 28, 24, 19, 13, 7, 1

Solid Angle Weight Calculation

Each patch's contribution to SVF is weighted by its solid angle. The discretized computation in skyview.rs:

n = 90
common_w_factor = (1 / (2π)) × sin(π / (2n))
steprad_iso = (360 / azimuth_patches) × (π / 180)
steprad_aniso = (360 / azimuth_patches_aniso) × (π / 180)

sin_term_sum = Σ sin(π(2a - 1) / (2n))    for a in annulino_start..=annulino_end

weight_iso = steprad_iso × common_w_factor × sin_term_sum
weight_aniso = steprad_aniso × common_w_factor × sin_term_sum

Where:

  • azimuth_patches = number of azimuthal divisions in this altitude band (isotropic)
  • azimuth_patches_aniso = ceil(azimuth_patches / 2) (directional, covering half-hemisphere)
  • annulino_start..annulino_end = range of annulus indices within the altitude band

The isotropic weight is used for total SVF. The anisotropic weight is used for directional SVFs (N, E, S, W), with each directional component only accumulating patches whose azimuth falls within its 180° half-hemisphere.

SVF Accumulation Formula

SVF = Σ_patches (weight_iso × visibility_patch)

Where:

  • weight_iso = pre-computed isotropic solid angle weight for the patch
  • visibility_patch = 1 if patch center is unobstructed, 0 if blocked by DSM

Last Annulus Correction

A correction factor LAST_ANNULUS_CORRECTION = 3.0459e-4 is added during finalization to svf_south and svf_west only (and their vegetation variants, conditioned on vegdem2 == 0). It is not applied to the isotropic svf, svf_north, or svf_east.

The reason for this asymmetry: the zenith patch has a single azimuthal subdivision at azimuth = 0°. The directional bucketing rules place azimuth 0° in the north (270°–90°) and east (0°–180°) half-hemispheres, but not in south (90°–270°) or west (180°–360°). The correction adds back the missing zenith contribution to the south and west directional SVFs. The isotropic SVF accumulates the zenith patch regardless of azimuth, so no correction is needed there.

Directional SVF

Directional components split patches by azimuth half-hemisphere:

SVF_north = Σ (weight_aniso × visibility) for azimuth ∈ [270°, 360°) ∪ [0°, 90°)
SVF_east  = Σ (weight_aniso × visibility) for azimuth ∈ [0°, 180°)
SVF_south = Σ (weight_aniso × visibility) for azimuth ∈ [90°, 270°)
SVF_west  = Σ (weight_aniso × visibility) for azimuth ∈ [180°, 360°)

Inputs

Input Type Description
DSM 2D array (m) Digital Surface Model
CDSM 2D array (m) Canopy DSM for vegetation (optional)
TDSM 2D array (m) Trunk DSM for vegetation trunk zone (optional)
pixel_size float (m) Resolution
usevegdem bool Whether to account for vegetation in SVF
max_local_dsm_ht float (m) Maximum local DSM height, limits ray march steps
patch_option int (1-4) Patch resolution option. Default 2 (153 patches)
min_sun_elev_deg float (°) Minimum elevation for shadow rays. Default 3.0

Outputs

Output Type Description
svf 2D array (0-1) Overall sky view factor
svf_north 2D array (0-1) SVF from northern half-hemisphere
svf_east 2D array (0-1) SVF from eastern half-hemisphere
svf_south 2D array (0-1) SVF from southern half-hemisphere
svf_west 2D array (0-1) SVF from western half-hemisphere
svf_veg 2D array (0-1) SVF accounting for vegetation canopy
svf_veg_north 2D array (0-1) Vegetation SVF from northern half-hemisphere
svf_veg_east 2D array (0-1) Vegetation SVF from eastern half-hemisphere
svf_veg_south 2D array (0-1) Vegetation SVF from southern half-hemisphere
svf_veg_west 2D array (0-1) Vegetation SVF from western half-hemisphere
svf_veg_blocks_bldg_sh 2D array (0-1) SVF where vegetation overlaps building shadow
svf_veg_blocks_bldg_sh_north 2D array (0-1) Directional variant (north)
svf_veg_blocks_bldg_sh_east 2D array (0-1) Directional variant (east)
svf_veg_blocks_bldg_sh_south 2D array (0-1) Directional variant (south)
svf_veg_blocks_bldg_sh_west 2D array (0-1) Directional variant (west)
bldg_sh_matrix 3D array (uint8) Bitpacked building shadow per patch
veg_sh_matrix 3D array (uint8) Bitpacked vegetation shadow per patch
veg_blocks_bldg_sh_matrix 3D array (uint8) Bitpacked veg-blocks-bldg shadow per patch

The shadow matrices (3D, uint8 bitpacked) store per-patch shadow results for use by anisotropic sky radiation. The third dimension indexes over patches.

Properties

Range Properties

  1. SVF in range [0, 1]
  2. SVF = 0: no sky visible (e.g., inside building)
  3. SVF = 1: full hemisphere visible (open field)

  4. All intermediate values valid

  5. Directional SVF in range [0, 1]

  6. Each directional component (N, E, S, W) also bounded by [0, 1]

Geometric Properties

  1. Flat open terrain = SVF of 1
  2. No obstructions → full sky visibility

  3. Tolerance: SVF > 0.95 for truly flat DSM

  4. Deep canyon has low SVF

  5. Urban canyon with H/W ratio > 2 → SVF < 0.5

  6. H = building height, W = street width

  7. Taller obstacles reduce SVF

  8. Higher buildings nearby → lower ground-level SVF

  9. SVF decreases monotonically with obstacle height

  10. Rooftops have high SVF

  11. Building tops (local maxima) have SVF close to 1

  12. Only reduced if taller buildings nearby

  13. Building density reduces SVF

  14. More buildings → lower ground-level SVF
  15. SVF is a measure of urban density/openness

Symmetry Properties

  1. Symmetric obstacles give symmetric directional SVF
  2. Square courtyard center has equal N/E/S/W SVF
  3. Asymmetric buildings create asymmetric directional SVF

Directional SVF

Directional components split the sky into half-hemispheres:

         N (svf_north)
         |
    W ---+--- E
         |
         S

Used for calculating radiation from different sky directions, important for:

  • Anisotropic sky radiance (brighter near sun)
  • Wall orientation effects
  • Asymmetric shading

Vegetation Effects

Trees reduce SVF but not completely (light passes through canopy):

  • svf_veg: Sky view through vegetation canopy
  • Accounts for the pergola shadow heuristic (see shadows.md)
  • svf_veg >= svf (vegetation blocks less than buildings)

References

Primary UMEP Citation:

  • Lindberg F, Grimmond CSB, Gabey A, Huang B, Kent CW, Sun T, Theeuwes N, Järvi L, Ward H, Capel-Timms I, Chang YY, Jonsson P, Krave N, Liu D, Meyer D, Olofson F, Tan JG, Wästberg D, Xue L, Zhang Z (2018) "Urban Multi-scale Environmental Predictor (UMEP) - An integrated tool for city-based climate services." Environmental Modelling and Software 99, 70-87. doi:10.1016/j.envsoft.2017.09.020

SVF Algorithm:

  • Lindberg F, Holmer B, Thorsson S (2008) "SOLWEIG 1.0 - Modelling spatial variations of 3D radiant fluxes and mean radiant temperature in complex urban settings." International Journal of Biometeorology 52(7), 697-713.
  • Lindberg F, Grimmond CSB (2011) "The influence of vegetation and building morphology on shadow patterns and mean radiant temperatures in urban areas: model development and evaluation." Theoretical and Applied Climatology 105, 311-323.

Patch-Based Method:

  • Robinson D, Stone A (1990) "Solar Radiation Modelling in the Urban Context." Building and Environment 25(3), 201-209.