Figure:
HDRF
Let Ω′\Omega'Ω′ be the incident solid angle, Ω\OmegaΩ is leaving solid angle. Consider the BRDF of a Lamvertian target is 1π\frac{1}{\pi}π1, the BRF is 1. The HDRF of a target is defined as:
Rhem(Ω)=ΦrΦrlam=LrLrlam=∫2π−fr(Ω′,Ω)dE′∫2π−1πdE′=∫2π−fr(Ω′,Ω)∣μ∣I′dΩ′1π∫2π−I′∣μ′∣dΩ′=I1π∫2π−I′∣μ′∣dΩ′, μ>0.
\begin{aligned}
R^{hem}(\Omega)&=\frac{\Phi_r}{\Phi_r^{lam}}=\frac{L_r}{L_r^{lam}}\\&=\frac{\int_{2\pi-}f_r(\Omega',\Omega)dE'}{\int_{2\pi-}\frac{1}{\pi}dE'}=\frac{\int_{2\pi-}f_r(\Omega',\Omega)|\mu| I'd\Omega'}{\frac{1}{\pi}\int_{2\pi-}I'|\mu'| d\Omega'}\\
&= \frac{I}{\frac{1}{\pi}\int_{2\pi-}I'|\mu'| d\Omega'}, \ \ \mu>0.
\end{aligned}
Rhem(Ω)=ΦrlamΦr=LrlamLr=∫2π−π1dE′∫2π−fr(Ω′,Ω)dE′=π1∫2π−I′∣μ′∣dΩ′∫2π−fr(Ω′,Ω)∣μ∣I′dΩ′=π1∫2π−I′∣μ′∣dΩ′I, μ>0.
This is equivelent to:
HDRF=<I(Ω)>01π∫2π−<I(Ω′)>0∣Ω′⋅n1∣dΩ′
HDRF=\frac{<I(\Omega)>_0}{\frac{1}{\pi}\int_{2\pi-}<I(\Omega')>_0 |\Omega'\cdot n_1|d\Omega'}
HDRF=π1∫2π−<I(Ω′)>0∣Ω′⋅n1∣dΩ′<I(Ω)>0
Here,n1n_1n1 is the outward normal, <⋅><\cdot><⋅> denotes mean over upper surface δVt\delta V_tδVt , and we will ignore this notion for a simplification in remaining part of the blog. HDRF depends on atmosphere conditions. HDRF is the ratio of real radiance to the radiance reflected from a lambertian target of canopy upper surface .
HDRF has no unit.
BRF
If no atmosphere, *i.e.,*incident solar radiation at upper canopy boundary δVt\delta V_tδVt is a parallel beam of light, the HDRF become BRF:
BRF=I(Ω)1πI(Ω′)∣Ω′⋅n1∣=I(Ω)ILam
BRF=\frac{I(\Omega)}{\frac{1}{\pi}I(\Omega')|\Omega'\cdot n_1|}=\frac{I(\Omega)}{I_{Lam}}
BRF=π1I(Ω′)∣Ω′⋅n1∣I(Ω)=ILamI(Ω)
Here, ILamI_{Lam}ILam is the radiance over upper surface of the canopy from a Lambertian target under the same illumination.
BRF does not depend on atmosphere conditions, only varies with Ω\OmegaΩ and Ω′\Omega'Ω′.
BRF has no unit.
BRDF
BRDF describes the scattering of a parallel beam of incident radiation from one direction into another direction. However, the denominator of BRDF is the incident flux, not the radiance.
BRDF=I(Ω)∫2π+ILam∣μ∣dΩ=I(Ω)ILamπ=I(Ω)ELam
BRDF=\frac{I(\Omega)}{\int_{2\pi+}I_{Lam}|\mu| d\Omega}=\frac{I(\Omega)}{I_{Lam}\pi}=\frac{I(\Omega)}{E_{Lam}}
BRDF=∫2π+ILam∣μ∣dΩI(Ω)=ILamπI(Ω)=ELamI(Ω)
From the second equality, we could get that π⋅BRDF=BRF\pi \cdot BRDF=BRFπ⋅BRDF=BRF.
BRDF has unit sr−1sr^{-1}sr−1.
BHR
The BHR is defined as mean irradiance exitance to incident irradiance:
A=∫2π+I(Ω)∣μ∣dΩ∫2π−I(Ω′)∣μ′∣dΩ′
A=\frac{\int_{2\pi+}I(\Omega)|\mu|d\Omega}{\int_{2\pi-}I(\Omega')|\mu'|d\Omega'}
A=∫2π−I(Ω′)∣μ′∣dΩ′∫2π+I(Ω)∣μ∣dΩ
BHR has no unit.
DHR
If no atomophere, BHR become DHR:
DHR=∫2π+I(Ω)∣μ∣dΩILam∣μ′∣
DHR=\frac{\int_{2\pi+}I(\Omega)|\mu|d\Omega}{I_{Lam}|\mu'|}
DHR=ILam∣μ′∣∫2π+I(Ω)∣μ∣dΩ
DHR has no unit.
For Lambertian Surface and no atmosphere
HDRF becomes:
HDRF=<I(Ω)>01π∫2π−<I(Ω′)>0∣Ω′⋅n1∣dΩ′=πIM
HDRF=\frac{<I(\Omega)>_0}{\frac{1}{\pi}\int_{2\pi-}<I(\Omega')>_0 |\Omega'\cdot n_1|d\Omega'}=\frac{\pi I}{M}
HDRF=π1∫2π−<I(Ω′)>0∣Ω′⋅n1∣dΩ′<I(Ω)>0=MπI
DHR becomes
DHR=∫2π+I(Ω)∣μ∣dΩILam∣μ′∣=πIM
DHR=\frac{\int_{2\pi+}I(\Omega)|\mu|d\Omega}{I_{Lam}|\mu'|}=\frac{\pi I}{M}
DHR=ILam∣μ′∣∫2π+I(Ω)∣μ∣dΩ=MπI
BHR becomes
A=I∫2π+∣μ∣dΩ∫2π−I(Ω′)∣μ′∣dΩ′=IπM
A=\frac{I\int_{2\pi+}|\mu|d\Omega}{\int_{2\pi-}I(\Omega')|\mu'|d\Omega'}=\frac{I\pi}{M}
A=∫2π−I(Ω′)∣μ′∣dΩ′I∫2π+∣μ∣dΩ=MIπ
BRF becomes:
BRF=I(Ω)1πI(Ω′)∣Ω′⋅n1∣=πIM
BRF=\frac{I(\Omega)}{\frac{1}{\pi}I(\Omega')|\Omega'\cdot n_1|}=\frac{\pi I}{M }
BRF=π1I(Ω′)∣Ω′⋅n1∣I(Ω)=MπI
So they are the same.