Radiation

Shortwave (solar) and longwave (thermal) radiation calculations.

Primary References:

• Lindberg et al. (2008) Sections 2.4-2.6
• Jonsson et al. (2006) - Sky emissivity formulation
• Reindl et al. (1990) - Diffuse fraction correlations
• Perez et al. (1993) - Anisotropic sky luminance distribution

Overview

Radiation at any point comes from six directions (up, down, and four cardinal sides), split into shortwave (K) and longwave (L) components:

Total radiation = Shortwave (K) + Longwave (L)
                = (Kdown + Kup + Kside) + (Ldown + Lup + Lside)

Source Code Locations

Radiation logic is spread across multiple Rust files:

• rust/src/perez.rs — Perez anisotropic sky model, clearness bins, luminance distribution
• rust/src/sky.rs — Isotropic sky radiation, cylindric wedge luminance
• rust/src/patch_radiation.rs — Per-patch sky emissivity (Martin & Berdahl band model)
• rust/src/emissivity_models.rs — Sky emissivity computation
• rust/src/pipeline.rs — Kdown, Kup, Ldown assembly from component outputs

Shortwave Radiation (K)

Solar radiation, wavelengths ~0.3-3 μm.

Diffuse Fraction (Preprocessing Step)

Global radiation (G) is split into direct (I) and diffuse (D) using the Reindl et al. (1990) correlation.

Reference: Reindl DT, Beckman WA, Duffie JA (1990) "Diffuse fraction correlations." Solar Energy 45(1):1-7.

Important: In this codebase, diffuse fraction splitting is performed as a preprocessing step (via clearnessindex_2013b and diffusefraction in the upstream UMEP Python). The Rust pipeline receives pre-split direct_rad (rad_i) and diffuse_rad (rad_d) as inputs. The Reindl model is documented here for reference but is not computed within the Rust pipeline.

The model uses piecewise correlations based on clearness index (Kt):

G = I + D
Kt = G / I0_et    (clearness index = ratio to extraterrestrial)

If Kt ≤ 0.3 (overcast):
    D/G = 1 - 0.232×Kt + 0.0239×sin(α) - 0.000682×Ta + 0.0195×RH

If 0.3 < Kt < 0.78 (partly cloudy):
    D/G = 1.329 - 1.716×Kt + 0.267×sin(α) - 0.00357×Ta + 0.106×RH

If Kt ≥ 0.78 (clear):
    D/G = 0.426×Kt - 0.256×sin(α) + 0.00349×Ta + 0.0734×RH

Where:

• α = solar altitude angle (radians)
• Ta = air temperature (°C)
• RH = relative humidity (fraction, 0-1)

Properties:

1. Clear sky: D/G ≈ 0.1-0.2 (mostly direct)
2. Overcast: D/G ≈ 0.9-1.0 (mostly diffuse)
3. D/G increases at low sun altitudes

Anisotropic Diffuse Sky (Perez Model)

For improved accuracy, diffuse radiation can use anisotropic sky luminance distribution.

Reference: Perez R, Seals R, Michalsky J (1993) "All-weather model for sky luminance distribution - Preliminary configuration and validation." Solar Energy 50(3):235-245.

The Perez model divides the sky into three components:

1. Isotropic background — uniform diffuse
2. Circumsolar brightening — enhanced near sun disk
3. Horizon brightening — enhanced near horizon

Sky Luminance Distribution

The relative luminance L at any sky element is given by:

L(θ,γ) = (1 + a×exp(b/cos(θ))) × (1 + c×exp(d×γ) + e×cos²(γ))

Where:

• θ = zenith angle of sky element (radians)
• γ = angular distance from sun (radians)
• a, b, c, d, e = coefficients from 8 sky clearness bins

Sky Clearness Categories

Sky clearness parameter ε determines coefficient bins:

Bin	ε Range	Description	Typical Condition
1	ε < 1.065	Very overcast	Heavy cloud cover
2	1.065-1.23	Overcast	Thick clouds
3	1.23-1.50	Cloudy	Medium clouds
4	1.50-1.95	Partly cloudy	Scattered clouds
5	1.95-2.80	Partly clear	Few clouds
6	2.80-4.50	Clear	Mostly clear
7	4.50-6.20	Very clear	Exceptionally clear
8	ε > 6.20	Extremely clear	Desert/high altitude

The clearness parameter ε is computed from (in rust/src/perez.rs):

ε = ((D + I) / D + 1.041 × zen³) / (1 + 1.041 × zen³)

Where zen is the solar zenith angle in radians. Note: the UTCI specification and some literature express this with θ_z in degrees using the coefficient 5.535×10⁻⁶. Both forms are equivalent: 1.041 × zen_rad³ = 5.535×10⁻⁶ × zen_deg³.

Bin-0 Special Equations (Robinson Extension)

For clearness bin 0 (most overcast), the c and d coefficients use different equations than bins 1-7:

c = exp((brightness × (c1 + c2 × zen))^c3) - 1
d = -exp(brightness × (d1 + d2 × zen)) + d3 + brightness × d4

This Robinson extension handles the overcast limiting case where the standard linear interpolation breaks down.

Anisotropic Sky Emissivity (Martin & Berdahl Band Model)

In anisotropic mode, per-patch sky emissivity is computed using the Martin & Berdahl (1984) band emissivity model (rust/src/patch_radiation.rs). This accounts for the angular dependence of atmospheric emission — the sky emits more longwave radiation near the horizon than at the zenith due to the longer atmospheric path length.

Implementation in SOLWEIG

The Rust implementation (rust/src/perez.rs):

1. Computes sky clearness bin from solar geometry and radiation
2. Retrieves coefficients (a, b, c, d, e) for the bin
3. Evaluates luminance L for each sky patch (altitude, azimuth)
4. Normalizes to ensure integration equals diffuse radiation
5. Returns patch luminance array for anisotropic radiation calculation

Kdown (Downwelling Shortwave)

Shortwave radiation reaching the ground from above:

Kdown = rad_i × shadow × sin(altitude)
      + drad
      + albedo_wall × (1 - svfbuveg) × (rad_g × (1 - f_sh) + rad_d × f_sh)

Where:

• rad_i × shadow × sin(altitude) = direct beam, blocked by shadows
• drad = diffuse sky component:
• Isotropic: drad = rad_d × svfbuveg
• Anisotropic: drad = ani_lum × rad_d (patch-weighted luminance)
• albedo_wall × (1 - svfbuveg) × (...) = wall-reflected shortwave
• f_sh = fraction of shadow (from SVF shadow matrices)

Properties:

1. Kdown proportional to SVF (for diffuse component)
2. Direct component blocked by shadows
3. Range: 0 to ~1000 W/m²

Kup (Reflected from Ground)

Shortwave reflected upward from ground surfaces:

ct = rad_d × svfbuveg + albedo_b × (1 - svfbuveg) × (rad_g × (1 - f_sh) + rad_d × f_sh)
kup = gvfalb × rad_i × sin(altitude) + ct × gvfalbnosh

Where:

• gvfalb = GVF albedo-weighted view factor (shadow-dependent, from gvf.md)
• gvfalbnosh = GVF albedo view factor (no shadow, geometric only)
• ct = combined diffuse + wall-reflected shortwave term

Note: gvfalb already contains albedo weighting — no additional multiplication by albedo is needed.

Properties:

1. Higher albedo → more reflection
2. Shaded areas reflect less (no direct component)
3. Range: 0 to ~200 W/m²

Kside (Direct + Reflected to Walls)

Shortwave reaching vertical surfaces, computed per cardinal direction (N, E, S, W):

Properties:

1. Depends on wall orientation relative to sun
2. South-facing walls receive more in Northern Hemisphere
3. Directional: Keast, Ksouth, Kwest, Knorth

Longwave Radiation (L)

Thermal radiation, wavelengths ~3-100 μm.

Sky Emissivity

Reference: Jonsson P, Eliasson I, Holmer B, Grimmond CSB (2006) "Longwave incoming radiation in the Tropics: Results from field work in three African cities." Theoretical and Applied Climatology 85:185-201.

Sky emissivity is computed from air temperature and humidity:

ea = 6.107 × 10^((7.5 × Ta) / (237.3 + Ta)) × (RH / 100)
msteg = 46.5 × (ea / Ta_K)
ε_sky = 1 - (1 + msteg) × exp(-√(1.2 + 3.0 × msteg))

Where:

• ea = water vapor pressure (hPa)
• Ta = air temperature (°C)
• Ta_K = air temperature (K)
• RH = relative humidity (%)

Typical values:

• Clear dry sky: ε_sky ≈ 0.60-0.75
• Clear humid sky: ε_sky ≈ 0.75-0.85
• Cloudy sky: ε_sky → 1.0

Ldown (Sky Longwave)

Thermal emission from atmosphere, computed in pipeline.rs::compute_ldown:

Ldown = (svf + svf_veg - 1) × esky × σ × Ta_K⁴
      + (2 - svf_veg - svf_aveg) × ewall × σ × Ta_K⁴
      + (svf_aveg - svf) × ewall × σ × (Ta + Tg_wall + 273.15)⁴
      + (2 - svf - svf_veg) × (1 - ewall) × esky × σ × Ta_K⁴

The four terms represent:

1. Sky emission through vegetation-free sky opening
2. Ambient-temperature wall emission (walls between vegetation and building canopy)
3. Heated wall emission (walls warmed by sun, between veg-blocking and building canopy)
4. Sky emission reflected off walls (non-absorbed sky radiation bounced from walls)

Cloud correction: When clearness index CI < 0.95:

Ldown_effective = Ldown_clear × (1 - c_cloud) + Ldown_cloudy × c_cloud
c_cloud = 1 - CI

Where Ldown_cloudy uses the same formula but with esky=1.0 for the sky term.

Properties:

1. Increases with humidity and cloud cover
2. Clear sky: Ldown ≈ 250-350 W/m²
3. Overcast: Ldown ≈ 350-450 W/m²

Lup (Ground Longwave)

Thermal emission from ground surface, provided by the GVF module after thermal delay smoothing:

Lup = TsWaveDelay(gvf_lup, ...)

Where gvf_lup is the ground-emitted longwave from the GVF integration (see gvf.md). The TsWaveDelay function applies exponential smoothing across timesteps to simulate thermal inertia (see ground_temperature.md).

Properties:

1. Increases with ground temperature
2. Hot asphalt can emit >500 W/m²
3. ε_ground typically 0.90-0.98

Lside (Wall Longwave)

Thermal emission from building walls, per cardinal direction:

Properties:

1. Sun-heated walls emit more
2. Directional: Least, Lsouth, Lwest, Lnorth
3. Important in urban canyons

Properties Summary

Shadow Effects

1. Shadows block direct shortwave only
2. Diffuse and longwave unaffected by shadows

3. Shaded areas still receive Kdown (diffuse), Ldown, Lup

4. Shadow reduces total K significantly

5. Sun to shade: ΔK ≈ 200-800 W/m² (depending on direct beam)

SVF Effects

1. Low SVF reduces sky radiation
2. Both Kdown and Ldown reduced

3. But Lside from walls increases

4. Urban canyon radiation balance

5. Lower Kdown, Ldown (less sky)
6. Higher Kup, Lside (more surfaces)

Temperature Effects

1. Hot surfaces increase longwave
2. Lup increases with ground temperature
3. Can dominate radiation budget on hot days

Typical Values

Component	Clear Day Noon	Shaded	Night
Kdown	100-200	100-200	0
Kup	50-150	30-100	0
Kside (sunlit)	200-600	0	0
I (direct)	600-900	0	0
Ldown	300-400	300-400	250-350
Lup	400-600	350-500	300-450
Lside	350-550	350-500	300-450

All values in W/m².

Implementation Notes

Clearness Index

Reference: Crawford TM, Duchon CE (1999) "An improved parameterization for estimating effective atmospheric emissivity for use in calculating daytime downwelling longwave radiation." Journal of Applied Meteorology 38:474-480.

The clearness index CI is pre-computed in the upstream UMEP Python preprocessing (clearnessindex_2013b) and supplied to the Rust pipeline as an input. It is not computed within the Rust pipeline.

Isotropic vs Anisotropic Mode

The model supports two diffuse radiation modes:

1. Isotropic (faster): Uniform diffuse sky. drad = rad_d × svfbuveg
2. Anisotropic (Perez, default): Non-uniform sky luminance, requires SVF shadow matrices from the SVF precomputation. Uses per-patch luminance weighting.

Use anisotropic mode when:

• High accuracy required near buildings
• Studying directional radiation effects
• SVF < 0.7 (urban canyons)

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

Radiation Model:

• 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.

Diffuse Fraction:

• Reindl DT, Beckman WA, Duffie JA (1990) "Diffuse fraction correlations." Solar Energy 45(1), 1-7.

Anisotropic Sky:

• Perez R, Seals R, Michalsky J (1993) "All-weather model for sky luminance distribution - Preliminary configuration and validation." Solar Energy 50(3), 235-245.

Sky Emissivity:

• Jonsson P, Eliasson I, Holmer B, Grimmond CSB (2006) "Longwave incoming radiation in the Tropics: Results from field work in three African cities." Theoretical and Applied Climatology 85, 185-201.

Anisotropic Emissivity:

• Martin M, Berdahl P (1984) "Characteristics of infrared sky radiation in the United States." Solar Energy 33(3-4), 321-336.