> ## Documentation Index
> Fetch the complete documentation index at: https://docs.plantpredict.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Bird Clear Sky Model

export const RayleighScattering = () => <Tooltip tip="Scattering of light by air molecules; responsible for blue sky and symmetric (50/50) forward/backward scatter.">
    Rayleigh scattering
  </Tooltip>;

export const ClosureEquation = () => <Tooltip tip="GHI = DNI × cos(θz) + DHI; relates the three horizontal irradiance components.">
    closure equation
  </Tooltip>;

export const AerosolOpticalDepth = () => <Tooltip tip="Measure of atmospheric turbidity due to aerosols; higher values indicate hazier conditions.">
    aerosol optical depth
  </Tooltip>;

export const PrecipitableWater = () => <Tooltip tip="Total water vapor in an atmospheric column, expressed as equivalent depth of liquid water (typically 0.5–5 cm).">
    precipitable water
  </Tooltip>;

export const SlantColumn = () => <Tooltip tip="Amount of an atmospheric constituent along the actual sun-to-surface path; equals column depth × air mass.">
    slant column
  </Tooltip>;

export const ColumnDepth = () => <Tooltip tip="Total amount of an atmospheric constituent in a vertical column, expressed as equivalent thickness at standard conditions.">
    column depth
  </Tooltip>;

export const AirMass = () => <Tooltip tip="Ratio of atmospheric path length to path length at zenith; equals 1 when sun is overhead.">
    air mass
  </Tooltip>;

export const ZenithAngle = () => <Tooltip tip="Angle between the local vertical and the sun (0° = overhead, 90° = horizon).">
    zenith angle
  </Tooltip>;

export const AtmosphericAttenuation = () => <Tooltip tip="Reduction of solar radiation as it passes through the atmosphere due to absorption and scattering.">
    atmospheric attenuation
  </Tooltip>;

export const Transmittance = () => <Tooltip tip="Fraction of solar radiation that passes through the atmosphere to reach the surface.">
    transmittance
  </Tooltip>;

export const DNI = () => <Tooltip tip="Direct Normal Irradiance: solar radiation received perpendicular to the sun's rays (direct beam only).">
    DNI
  </Tooltip>;

export const DHI = () => <Tooltip tip="Diffuse Horizontal Irradiance: solar radiation from the sky dome (excluding direct beam) on a horizontal surface.">
    DHI
  </Tooltip>;

export const GHI = () => <Tooltip tip="Global Horizontal Irradiance: total solar radiation on a horizontal surface (direct + diffuse).">
    GHI
  </Tooltip>;

## 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.

## Inputs

| Name                                          | Symbol        | Units   | Description                                                  |
| --------------------------------------------- | ------------- | ------- | ------------------------------------------------------------ |
| **Extraterrestrial Direct Normal Irradiance** | $DNI_{extra}$ | W/m²    | Solar irradiance at top of atmosphere                        |
| **Solar Zenith Angle**                        | $\theta_z$    | degrees | Angle between zenith and sun position                        |
| **Ground Albedo**                             | $\rho_g$      | —       | Surface reflectance for sky-ground coupling (0–1)            |
| **Atmospheric Pressure**                      | $P$           | hPa     | Local atmospheric pressure (for pressure-corrected air mass) |

***

## Outputs

| Name                                        | Symbol        | Units | Description                                                                    |
| ------------------------------------------- | ------------- | ----- | ------------------------------------------------------------------------------ |
| **Clear-Sky Global Horizontal Irradiance**  | $GHI_{clear}$ | W/m²  | Clear-sky total irradiance on horizontal surface including sky-ground coupling |
| **Clear-Sky Diffuse Horizontal Irradiance** | $DHI_{clear}$ | W/m²  | Clear-sky diffuse irradiance from sky dome                                     |
| **Clear-Sky Direct Normal Irradiance**      | $DNI_{clear}$ | W/m²  | Clear-sky beam irradiance perpendicular to sun                                 |

***

## 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 (<RayleighScattering />) 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.

### Atmospheric Constituents

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

| Constituent             | Symbol                    | Effect                    |
| ----------------------- | ------------------------- | ------------------------- |
| Air molecules           | $T_R$                     | Rayleigh Scattering       |
| Ozone                   | $T_{O_3}$                 | Absorption                |
| Mixed gases (CO₂, etc.) | $T_g$                     | Absorption                |
| Water vapor             | $T_W$                     | Absorption                |
| Aerosols                | $T_A$, $T_{AA}$, $T_{AS}$ | Absorption and Scattering |

### Transmittance Equations

The following transmittance equations are empirical fits from Bird & Hulstrom (1981). Most equations depend on <AirMass />, 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](../solar-geometry/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 <SlantColumn /> $x_O = U_O \times AM$, the product of <ColumnDepth /> 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 <SlantColumn /> $x_W = u_W \times AM$, the product of <PrecipitableWater /> (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 <AerosolOpticalDepth /> $\tau_A$, 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).

### 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).

### 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.

### 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 <ClosureEquation /> ($GHI = DNI \cos(\theta_z) + DHI$) because the Bird model calculates DHI from atmospheric scattering alone, requiring explicit treatment of ground-sky multiple reflections.

### High Zenith Angle Handling

The <ZenithAngle /> 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
* $GHI_{clear} = 0$

***

## 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. DOI: [10.2172/6510849](https://doi.org/10.2172/6510849)
* 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. DOI: [10.1016/S0038-092X(03)00195-6](https://doi.org/10.1016/S0038-092X\(03\)00195-6)