Bird Clear Sky Model

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Summary

The Bird Clear Sky Model computes theoretical clear-sky irradiance components (GHI, DHI, DNI) using empirical transmittance functions for atmospheric constituents. Developed by Bird and Hulstrom (1981), PlantPredict uses this model primarily for spectral correction calculations, where the clear sky index (ratio of measured GHI to clear-sky GHI) is an input to the spectral shift model. Clear-sky irradiance values are also included in prediction outputs for reference.

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Inputs

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Outputs

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Detailed Description

The Bird model calculates how extraterrestrial irradiance ($DNI_{extra}$) is affected by two processes as it passes through the atmosphere under clear-sky conditions:

• Absorption: Solar energy is absorbed by atmospheric gases (ozone, water vapor, mixed gases) and aerosols, converting radiation to heat—this energy is lost
• Scattering: Solar radiation is redirected by air molecules (Rayleigh scattering) and aerosols, removing it from the direct beam—this energy becomes diffuse irradiance

The model calculates a transmittance factor ($T$, ranging from 0 to 1) for each atmospheric constituent. Clear-sky DNI is the product of extraterrestrial irradiance and all transmittances. Clear-sky DHI is calculated from the scattered radiation. For aerosols, the model separates absorption ($T_{AA}$) from scattering ($T_{AS}$) because only scattered light contributes to DHI.

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Atmospheric Constituents

The following table summarizes the atmospheric constituents considered and whether they contribute to absorption, scattering, or both:

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Transmittance Equations

The following transmittance equations are empirical fits from Bird & Hulstrom (1981). Most equations depend on air mass, which represents the normalized path length through the atmosphere. The model calculates both relative air mass ($AM$) and pressure-corrected air mass ($AM'$) internally using the Bird-Hulstrom formula (see Air Mass for details). Air Molecules (Rayleigh Scattering): Air molecules scatter solar radiation, removing radiation from the direct beam and contributing to diffuse irradiance. The transmittance depends on the pressure-corrected air mass: $$T_R = \exp\left(-0.0903 \cdot AM'^{0.84} \cdot (1 + AM' - AM'^{1.01})\right)$$ The scattered fraction $(1 - T_R)$ contributes to DHI. Ozone: Ozone in the stratosphere absorbs UV radiation and some visible light. The transmittance depends on the ozone slant column $x_O = U_O \times AM$, the product of column depth and air mass. PlantPredict uses a fixed ozone column depth $U_O = 0.3$ cm: $$T_{O_3} = 1 - 0.1611 \cdot x_O \cdot (1 + 139.48 \cdot x_O)^{-0.3035} - \frac{0.002715 \cdot x_O}{1 + 0.044 \cdot x_O + 0.0003 \cdot x_O^2}$$ Mixed Gases: Uniformly mixed gases (primarily CO₂ and O₂) contribute minor absorption across the solar spectrum. The transmittance depends on pressure-corrected air mass: $$T_g = \exp(-0.0127 \cdot AM'^{0.26})$$ Water Vapor: Water vapor absorbs strongly in the near-infrared. The transmittance depends on the water vapor slant column $x_W = u_W \times AM$, the product of precipitable water (water vapor column depth) and air mass. PlantPredict uses a fixed precipitable water $u_W = 1.5$ cm: $$T_W = 1 - \frac{2.4959 \cdot x_W}{(1 + 79.034 \cdot x_W)^{0.6828} + 6.385 \cdot x_W}$$ Aerosols: Aerosols (dust, haze, pollution) both absorb and scatter solar radiation. The transmittance depends on broadband aerosol optical depth $\tau_A$, calculated from spectral values $t_{A}$ at 380 nm and 500 nm: $$\tau_A = 0.2758 \cdot t_{A,380} + 0.35 \cdot t_{A,500}$$ PlantPredict uses fixed values $t_{A,380} = 0.1$ and $t_{A,500} = 0.15$, representing typical rural conditions. For DNI, both absorption and scattering remove radiation from the direct beam: $$T_A = \exp\left(-\tau_A^{0.873} \cdot (1 + \tau_A - \tau_A^{0.7088}) \cdot AM^{0.9108}\right)$$ For DHI calculations, absorption and scattering must be separated because scattered light contributes to diffuse irradiance while absorbed light is lost. The aerosol absorption transmittance ($T_{AA}$) and aerosol scattering transmittance ($T_{AS}$) are: $$T_{AA} = 1 - K_1 \cdot (1 - AM + AM^{1.06}) \cdot (1 - T_A)$$ $$T_{AS} = \frac{T_A}{T_{AA}}$$ where $K_1 = 0.1$ is the aerosol absorptance fraction (10% of attenuated light is absorbed, 90% is scattered).

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Direct Normal Irradiance

Clear-sky DNI is the product of extraterrestrial irradiance and all transmittances: $$DNI_{clear} = DNI_{extra} \times 0.9662 \times T_R \times T_{O_3} \times T_g \times T_W \times T_A$$ The factor 0.9662 is an empirical calibration constant from Bird & Hulstrom (1981).

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Diffuse Horizontal Irradiance

Clear-sky DHI is calculated from scattered radiation, with contributions from Rayleigh scattering by air molecules and forward scattering by aerosols: $$\begin{aligned} DHI_{clear} &= DNI_{extra} \cos(\theta_z) \times 0.79 \times T_{O_3} \times T_g \times T_W \times T_{AA} \\ &\quad \times \frac{0.5(1 - T_R) + b_A(1 - T_{AS})}{1 - AM + AM^{1.02}} \end{aligned}$$ The numerator in the last term represents the two diffuse sources:

• $0.5(1 - T_R)$: Half of Rayleigh-scattered light (the other half scatters upward)
• $b_A(1 - T_{AS})$: Forward-scattered aerosol light, where $b_A = 0.84$ is the forward scatter ratio (84% of light scattered by aerosols continues downward)

The factor 0.79 is an empirical calibration constant.

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Global Horizontal Irradiance

GHI combines direct and diffuse components, accounting for multiple reflections between the ground and sky: $$GHI_{clear} = \frac{DNI_{clear} \cos(\theta_z) + DHI_{clear}}{1 - \rho_g \rho_s}$$ where the sky albedo $\rho_s = 0.0685 + (1 - b_A)(1 - T_{AS})$ represents the fraction of upwelling radiation reflected back down by the atmosphere. This differs from the standard closure equation ($GHI = DNI \cos(\theta_z) + DHI$) because the Bird model calculates DHI from atmospheric scattering alone, requiring explicit treatment of ground-sky multiple reflections.

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High Zenith Angle Handling

The zenith angle is limited to 87.9° to stay within the valid range of the air mass model. If $\theta_z \geq 87.9°$:

• $DNI_{clear} = 0$
• $DHI_{clear} = 0.001$ W/m²
• $GHI_{clear} = 0$

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References

• Bird, R. E., & Hulstrom, R. L. (1981). A simplified clear sky model for direct and diffuse insolation on horizontal surfaces. SERI/TR-642-761, Solar Energy Research Institute.
• Gueymard, C. A. (2003). Direct solar transmittance and irradiance predictions with broadband models. Part I: Detailed theoretical performance assessment. Solar Energy, 74(5), 355–379.