Figure:

HDRF

Let ${\Omega }^{\prime }$ be the incident solid angle, $\Omega$ is leaving solid angle. Consider the BRDF of a Lamvertian target is $\frac{1}{\pi }$, the BRF is 1. The HDRF of a target is defined as:
$$\begin{array}{rl}{R}^{hem}(\Omega )& =\frac{{\Phi }_r}{{\Phi }_r^{lam}}=\frac{L_r}{L_r^{lam}}\\ & =\frac{{\int }_{2\pi -}{f}_r({\Omega }^{\prime },\Omega )d{E}^{\prime }}{{\int }_{2\pi -}\frac{1}{\pi }d{E}^{\prime }}=\frac{{\int }_{2\pi -}{f}_r({\Omega }^{\prime },\Omega )|\mu |I^{\prime }d{\Omega }^{\prime }}{\frac{1}{\pi }{\int }_{2\pi -}{I}^{\prime }|{\mu }^{\prime }|d{\Omega }^{\prime }}\\ & =\frac{I}{\frac{1}{\pi }{\int }_{2\pi -}{I}^{\prime }|{\mu }^{\prime }|d{\Omega }^{\prime }},\mu >0.\end{array}$$
This is equivelent to:
$$HDRF=\frac{<I(\Omega ){>}_0}{\frac{1}{\pi }{\int }_{2\pi -}<I({\Omega }^{\prime }){>}_0|{\Omega }^{\prime }\cdot n_1|d{\Omega }^{\prime }}$$

Here,$n_1$ is the outward normal, $<\cdot >$ denotes mean over upper surface $\delta V_t$​ , 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 $\delta V_t$ is a parallel beam of light, the HDRF become BRF:
$$BRF=\frac{I(\Omega )}{\frac{1}{\pi }I({\Omega }^{\prime })|{\Omega }^{\prime }\cdot n_1|}=\frac{I(\Omega )}{I_{Lam}}$$
Here, $I_{Lam}$​​ 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 $\Omega$ and ${\Omega }^{\prime }$​.

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.
$$BRDF=\frac{I(\Omega )}{{\int }_{2\pi +}{I}_{Lam}|\mu |d\Omega }=\frac{I(\Omega )}{I_{Lam}\pi }=\frac{I(\Omega )}{E_{Lam}}$$
From the second equality, we could get that $\pi \cdot BRDF=BRF$​.

BRDF has unit $s{r}^{-1}$.

BHR

The BHR is defined as mean irradiance exitance to incident irradiance:
$$A=\frac{{\int }_{2\pi +}I(\Omega )|\mu |d\Omega }{{\int }_{2\pi -}I({\Omega }^{\prime })|{\mu }^{\prime }|d{\Omega }^{\prime }}$$
BHR has no unit.

DHR

If no atomophere, BHR become DHR:
$$DHR=\frac{{\int }_{2\pi +}I(\Omega )|\mu |d\Omega }{I_{Lam}|{\mu }^{\prime }|}$$
DHR has no unit.

For Lambertian Surface and no atmosphere

HDRF becomes:
$$HDRF=\frac{<I(\Omega ){>}_0}{\frac{1}{\pi }{\int }_{2\pi -}<I({\Omega }^{\prime }){>}_0|{\Omega }^{\prime }\cdot n_1|d{\Omega }^{\prime }}=\frac{\pi I}{M}$$
DHR becomes
$$DHR=\frac{{\int }_{2\pi +}I(\Omega )|\mu |d\Omega }{I_{Lam}|{\mu }^{\prime }|}=\frac{\pi I}{M}$$
BHR becomes
$$A=\frac{I{\int }_{2\pi +}|\mu |d\Omega }{{\int }_{2\pi -}I({\Omega }^{\prime })|{\mu }^{\prime }|d{\Omega }^{\prime }}=\frac{I\pi }{M}$$
BRF becomes:
$$BRF=\frac{I(\Omega )}{\frac{1}{\pi }I({\Omega }^{\prime })|{\Omega }^{\prime }\cdot n_1|}=\frac{\pi I}{M}$$
So they are the same.