**Page updated:**
April 15, 2020 **Author:** Curtis Mobley

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# The BRDF

We next consider how light is reﬂected by opaque surfaces, such as a sandy sea bottom. Scientists in many diﬀerent ﬁelds including astronomy, geology, agronomy, the paint industry, camouﬂage technology, and remote sensing have studied how surfaces reﬂect light. Unfortunately, diﬀerent ﬁelds often use diﬀerent measures of ”reﬂectance,” and they all have their own terminology and notation even when they are measuring the same physical quantity. There are many opportunities for losing factors of $\pi $ and cosines of angles, and it is sometimes nearly impossible to ﬁgure out exactly what is being discussed when reading a paper. This page gives an overview of the deﬁnitions, terminology, and notation as needed for optical oceanography and remote sensing.

For the most part, the deﬁnitions and terminology used here are given in Hapke (1993), which is a good introductory textbook on reﬂectance, and in Nicodemus, et al. (1977) (referenced here as NBS160). NBS160 is a National Bureau of Standards document that discusses the measurement of reﬂectance in great detail and is the authoritative document on the subject. However, we have changed some notation to correspond to what is commonly used in optical oceanography. Table 1 at the end of this page compares the notation used in these books.

For convenience, let the ”surface” reﬂecting the light be a horizontal plane. This can be a physical surface such as a sandy ocean bottom, or it can be simply a particular depth in the water column, say at 1 m above a sea grass bed or at 100 m in optically deep mid-ocean water. To conform to NBS160, we’ll use subscript i to denote incident and r to denote reﬂected. In the oceanographic setting of a horizontal bottom, the light incident onto the surface is traveling downward, and the light reﬂected by the surface is traveling upward. Thus we sometimes use subscript d for downward (incident) and u for upward (reﬂected) when necessary to conform to common oceanographic usage.

In nature, light is usually incident onto a surface from all directions, and some of the incident light gets reﬂected by the surface into all directions. Therefore, to completely understand the optical properties of a surface, it is necessary to know how the surface reﬂects light going in any incident direction into any reﬂected direction.

Figure 1 shows the geometry used to describe reﬂectance from a surface. A cartesian (x,y,z) coordinate system is chosen with the surface lying in the x-y plane and with the z axis normal (upward in our case) to the surface, an element of which is shown in aqua. There is a collimated light source, which provides the incident light, in direction $\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)$; and there is a detector, which receives the reﬂected light, located at the viewing direction $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$. Surface optical properties usually depend on the wavelength $\lambda $, so the complete description of the reﬂectance properties of a surface will be a function (the BRDF) of ﬁve variables: ${\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r},\lambda $. To make our equations as simple as possible, we drop the $\lambda $, but keep in mind that everything discussed below depends on wavelength.

For oceanography, it is often reasonable to assume that the surface is azimuthally isotropic, which means that its reﬂectance properties depend on the diﬀerence of ${\varphi}_{i}$ and ${\varphi}_{r}$. (This would not the case for long parallel ripples on a sandy bottom, for example.) The specular direction is the direction that a level mirror surface would reﬂect light: $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)=\left({\mathit{\theta}}_{i},{\varphi}_{i}+180\right)$. The retroreﬂection direction is the direction of exact backscatter: $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)=\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)$. The angle $\xi $ between the source and detector is called the phase angle; it is computed from

[Comment: If the source is the sun and the surface is the moon and the earth is the detector, then the phase angle determines the phase of the moon as seen from the earth. This is the historical origin of the term ”phase function” for the function that describes the angular pattern of scattered light. The scattering angle $\psi $ as used in radiative transfer theory is the complement of the phase angle: $\psi =180-\xi $.]

Great care and precise language must be used when talking about reﬂectance. For example, ”reﬂectance” should be preceded by two adjectives: the ﬁrst describes the source and the second the detector. Thus we have

- the directional-hemispherical reﬂectance: tells how much light is reﬂected from a particular direction $\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)$ into the hemisphere of all upward directions
- the hemispherical-directional reﬂectance: tells how much light is reﬂected from all downward directions into a particular direction $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$. The remote-sensing reﬂectance ${R}_{rs}={L}_{w}\u2215{E}_{d}$ used in optical oceanography is a hemispherical-directional reﬂectance.
- the hemispherical-hemispherical (or bi-hemispherical) reﬂectance: tells how much light is reﬂected from all downward directions into all upward directions. The irradiance reﬂectance $R={E}_{u}\u2215{E}_{d}$ used in optical oceanography is a bi-hemispherical reﬂectance.

We now deﬁne the bi-directional (i.e., directional-directional) reﬂectance distribution function (BRDF), which tells us everything we need to know about how a surface reﬂects light. The following discussion is based on NBS160, which treats these matters in great detail.

Conceptually, we think about a light beam traveling in a particular direction $\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)$ being reﬂected into another particular direction $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$. But since any source has some ﬁnite divergence, and any detector has some ﬁnite ﬁeld of view, we can associate small solid angles $d{\Omega}_{i}$ and $d{\Omega}_{r}$ with the incident and reﬂected beams, respectively. The radiance of the incident beam is ${L}_{i}\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)$, and ${L}_{r}\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$ is the reﬂected radiance. These quantities are shown in Fig. 2, which is a redrawn version of Fig. 1.

Our goal is to deﬁne an inherent optical property that tells us how the reﬂective properties of the surface vary with incident and reﬂected directions (and wavelength). Therefore, consider a measurement in which we hold the direction of the detector in Fig. 2 constant while we vary the direction of the source. The BRDF is then deﬁned as

Note that if only the magnitude of the incident radiance changes, the reﬂected radiance will change proportionately, and the BRDF will remain unchanged. However, if the direction of the incident or reﬂected beams changes while holding all else constant, the BRDF will in general change.

Equation (2) allows an easy transition to radiative transfer theory. Suppose we want to compute the total radiance heading upward in direction $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$ owing to light incident onto the surface from all directions. We then rewrite (2) as

and then integrate over all incident directions to get the total reﬂected radiance in direction $\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)$:

$$\begin{array}{lll}\hfill {L}_{r}\left({\mathit{\theta}}_{r},{\varphi}_{r}\right)\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}{\int}_{2{\pi}_{i}}\phantom{\rule{0.3em}{0ex}}{L}_{i}\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)\phantom{\rule{0.3em}{0ex}}BRDF\left({\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r}\right)\phantom{\rule{0.3em}{0ex}}cos{\mathit{\theta}}_{i}\phantom{\rule{0.3em}{0ex}}d{\Omega}_{i}\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \equiv & \phantom{\rule{1em}{0ex}}{\int}_{2{\pi}_{i}}\phantom{\rule{0.3em}{0ex}}{L}_{i}\left({\mathit{\theta}}_{i},{\varphi}_{i}\right)\phantom{\rule{0.3em}{0ex}}r\left({\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r}\right)\phantom{\rule{0.3em}{0ex}}d{\Omega}_{i}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\end{array}$$

This last equation (5) is exactly what is seen (with slightly diﬀerent notation) in Light and Water (1994) Eq. (4.3), where $r\left({\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r}\right)$ is called the radiance reﬂectance function. Clearly, $r\left({\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r}\right)=BRDF\left({\mathit{\theta}}_{i},{\varphi}_{i},{\mathit{\theta}}_{r},{\varphi}_{r}\right)cos{\mathit{\theta}}_{i}$, and the two functions are equivalent ways of describing a surface. In radiative transfer theory irradiances are measured on surfaces normal to the direction of light propagation, whereas actual irradiance measurements are made on the surface of interest. The $cos{\mathit{\theta}}_{i}$ factor in Eq. (4) just projects the incident beam irradiance onto the horizontal surface. This is one of those places where it is easy to lose a cosine factor when comparing an observational paper and a theory paper. Also, some investigators add a factor of $\pi \phantom{\rule{1em}{0ex}}sr$ to the numerator of Eq. (2) and deﬁne the BRDF as a nondimensional quantity, although this is non-standard. [However, this is how the MODTRAN atmospheric radiative transfer model deﬁnes its BRDFs for various types of earth surfaces that form the bottom boundary of the atmosphere.] Finally, note that the BRDF is a reﬂectance per unit solid angle; it can have any non-negative value. It is only when the BRDF is integrated over solid angle to get, for example, an irradiance reﬂectance that the resulting reﬂectance is bounded by one.

It is emphasized that the BRDF completely describes the net eﬀect of everything that happens on or below the surface where it is measured. For example, if the BRDF is measured in the water column 1 m above a sea grass bed, then all the eﬀects of the light interacting with the grass, sediments, and water below the 1 m surface are accounted for in this BRDF. Knowing the BRDF on this imaginary surface would, for example, allow HydroLight to compute the radiance distribution in the region above the depth where the BRDF was measured. Predicting or computing the BRDF of the grass and sediments is, however, very diﬃcult and requires understanding and modeling all of the extremely complicated interactions of light with the grass and sediment particles.

[Comment: This is how HydroLight models inﬁnitely deep, homogeneous water without actually solving the radiative transfer equation to extreme depth. The BRDF of an inﬁnitely deep, homogeneous layer of water with known inherent optical properties can be found analytically (as in Light and Water Section 9.5). Thus, when HydroLight simulates inﬁnitely deep water, it ﬁrst computes the BRDF of the inﬁnitely deep water below the maximum depth ${z}_{max}$ of interest, and it then uses that BRDF at ${z}_{max}$ just as though there were an actual physical bottom at ${z}_{max}$.]

Finally, there is an important reciprocity theorem about what happens if the positions if the source and detector are interchanged. It states simply that

If you measure or deﬁne a BRDF that does not obey Eq. (6), then it is simply wrong.

Quantity | This page | Light and Water | Hapke (1993) | NBS160 |

radiance | L | L | I | L |

irradiance | E | E | J | E |

single-scattering albedo | ${\omega}_{o}$ | ${\omega}_{o}$ | w | — |

scattering angle | $\psi $ | $\psi $ | $\mathit{\theta}$ | — |

| g | g | $\xi $ | — |

phase angle | $\xi $ | — | g | — |

incident polar angle | ${\mathit{\theta}}_{i}$ | ${\mathit{\theta}}^{\prime}$ | i | ${\mathit{\theta}}_{i}$ |

reﬂected polar angle | ${\mathit{\theta}}_{r}$ | $\mathit{\theta}$ | e | ${\mathit{\theta}}_{r}$ |

incident azimuthal angle | ${\varphi}_{i}$ | ${\varphi}^{\prime}$ | set to 0 | ${\varphi}_{i}$ |

reﬂected azimuthal angle | ${\varphi}_{r}$ | $\varphi $ | $\psi $ | ${\varphi}_{r}$ |

solid angle | $\Omega $ | $\Omega $ | $\Omega $ | $\omega $ |

BRDF | BRDF | $r\u2215cos{\mathit{\theta}}_{i}$ | BRDF | ${f}_{r}$ |

irradiance reﬂectance | R | R | r | $\rho $ |