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If we could always measure or model the complete BRDF of a surface our radiative transfer problems would be solved. However, measuring even a partial BRDF for a few values of is a difficult and tedious task in the laboratory and is almost never attempted in the ocean. Of course, everyone wants to have some easily made (compared to a BRDF) measure of surface reflectance that, with appropriate assumptions, can be used to describe the optical properties of the surface. This leads us to various other reflectances and quantities derived from the BRDF.
AlbedosThere are many definitions of "albedo." Hapke (1993) defines bolometric, Bond, geometric, hemispherical, normal, physical, plane, singlescattering, and spherical albedos, as well as an albedo factor. Not a single one of these albedos corresponds to how albedo is defined in Light and Water (1994) (page 193). Fortunately, oceanographers do not have to deal with all of these albedos, but we should clarify a point that can cause confusion when reading papers on reflectance (which sometimes just say, "...and the albedos is..." without telling you which one they are using). Optical oceanographers generally think of the albedo as being the ratio of the upwelling plane irradiance to the downwelling plane irradiance, for whatever conditions of incident lighting you have in nature at the time of measurement. (This is how the albedo is defined in Light and Water.) This is what you need to know to compute an energy balance in the ocean, for example. Thus the oceanographers' albedo is the same as the irradiance reflectance . Many surfacereflectance scientists (e.g., in the paint industry) like to define their albedos and reflectances in terms of isotropic illumination of the surface, i.e., the incident radiance ) is a constant independent of . For isotropic incident radiance, the general equation for the irradiance reflectance (see the page on Lambertian BRDFs)
reduces to just
This quantity is called the Bond or spherical albedo, or the spherical or bihemispherical reflectance, or just the albedo, depending on the author's preference. Note that this A is not equal to unless the incident lighting is isotropic (which never occurs in nature) or unless the surface is Lambertian (which never occurs in nature). For a Lambertian surface, , where is the reflectivity of the surface. The same convention of assuming isotropic illumination of the surface is often used when defining other reflectances, e.g., the hemisphericaldirectional reflectance. Note that the convention of using isotropic illumination when defining albedos and reflectances is not necessarily bad: it removes a complicating factorvariable incident lightingfrom the discussion of surface properties. However, oceanographers cannot control their incident lighting; they have to live with whatever incident radiance nature gives them.
The Irradiance Reflectance vs. The BiHemispherical ReflectanceAs already noted, the oceanographers' albedo or irradiance reflectance as given by Eq. (1) is a bihemispherical reflectance, but it is not the bihemispherical reflectance as defined in books such as Hapke's, because the oceanographer's uses the actual incident radiance distribution in Eq. (1), rather than an isotropic incident radiance.
The RemoteSensing ReflectanceThe oceanographers' remotesensing reflectance
has the same units as the BRDF, but they are not the same thing. Note in particular that uses the downwelling radiance from all directions (as contained in ), whereas the incident radiance in the definition of the BRDF is in a collimated beam. [Also, some people call this ratio the "remotesensing reflectance" only if the measurement is being made just above the sea surface and is the waterleaving radiance (the total upward radiance minus the surfacereflected sky radiance). If measured in water, the ratio of upwelling radiance to downwelling irradiance is then called the "remotesensing ratio."]
The Reflectance Factor and the Radiance FactorThe reflectance factor REFF (also called the reflectance coefficient) is defined as the ratio of the BRDF of a surface to that of a perfectly diffuse surface under the same conditions of illumination and observation. "Perfectly diffuse" means a Lambertian surface with . Thus
The radiance factor RADF is defined as the reflectance factor for normal illumination, i.e., for . Thus
The Fresnel ReflectanceThe Fresnel reflectance is the reflectance of a perfectly smooth surface between two media of different indices of refraction . Formulas for are given in almost any optics text, and in Light and Water (1994, page 157). Figure 1 gives for unpolarized light incident onto either side of an airwater surface; the water has a real index of refraction (relative to the air) of .
Note that the Fresnel reflectance describes the reflectance of the surface itself. The irradiance and remotesensing reflectances, and , when measured just above the sea surface describe the reflectance of the sea surface plus the water beneath the surface. can be combined with Dirac delta functions to create a BRDF. Consider the BRDF
Inserting Eq. (1) into the general equation for reflected radiance (Eq. (4) of the BRDF page),
gives
This last equation is the form usually seen in the definition of the Fresnel reflectance as being the ratio of reflected to incident (ir)radiances for angles related by the law of reflection.

