Bidirectional reflectance distribution function (BRDF)

Figure 2.1 illustrates the tendency o f natural surfaces tend to reflect incident radiation anisotropically. This behaviour is formally described by the (spectral) bidirectional reflectance distribution function (BRDF) (Nicodemus, 1970; Nicodemus et a l, 1997; Hapke, 1981). The BRDF o f a small surface area ôA (at a particular wavelength o f (non-polarised) illuminating radiance) is defined as the ratio o f the incremental radiance leaving the surface through an infinitesimal solid angle in the direction defined by the viewing vector, Q(0v, ^v) (where 0v, are the viewing zenith and azimuth angles) to the incremental irradiance from direction defined by the illumination vector, Q ’(0/, 4)/) (where 0„ (|), are the illumination zenith and azimuth angles). This is expressed in equation 2.2 (ignoring any dependence o f BRDF on wavelength, X)

W here dLc is the incremental radiance reflected from the surface into the differential solid angle in the viewing direction Q (Wm'^sr ') (Nicodemus et a l, 1977; M artonchik et a l,

viewer incid en t d iffu se radiation direct irradiance (ÔE/) vector Q ' exitant so lid

angle 5Q incident so lid angle ÔQ'

su rface area 5 A surface tangent vector

Figure 2.2 Configuration o f viewing and illumination vectors in the viewing hemisphere, with respect to an element o f surface area, ÔA.

2000); dE; is the incremental irradiance (Wm'^sr"') arriving from the illumination direction, Q ’ i.e. dE. = L.(Q')cos<9. sm(j)^d6-d(p^. Figure 2.2 shows the configuration.

Equation 2.2 only deals with two o f the domains o f information described in the expression for measured reflectance given in equation 2.1, but for exploring spectral directional reflectance these are the important ones. BRDF as defined in equation 2.2 is a fundamental property o f the surface, describing the intrinsic surface reflectance. However, it is defined only for infinitesimal viewing and illumination solid angles and an infinitesimal wavelength interval. Consequently it cannot be directly measured. In EO applications, illumination is typically over a hemisphere with both direct and diffuse sources (solar illumination, and sky radiance respectively). Viewing is typically over some finite sensor instantaneous field-of-view (IFOV), defined by the sensor optics and geometry, with a spectral response over some finite wavelength interval rather than at some discrete value o f X.

In practice, it is assumed that the BRDF can be retrieved with some level o f uncertainty, from radiance measurements over the IFOV using a collimated beam light source (e.g. laser or direct sunlight). In this case, BRDF is the limit case o f biconical reflectance measurement. To overcome the problem that the BRDF is a non-measurable property, we define a bidirectional reflectance factor (BRE) p (1 0 , Q ’), which is the ratio o f radiance leaving the surface in a finite solid angle in the viewing direction Q to the

radiance from a perfect Lambertian reflector under the same illumination conditions as the target, into the same finite solid angle i.e.

where Le is the radiance exitant from the surface (W m '^sr’); LLambertian is the radiance from a perfect Lambertian reflector (Wm’^sr'^); Lsky.sun are the sky and sun radiance distributions (Wm'^sr '). It can be seen that the BRF is dimensionless.

As a result o f the incident irradiance and exitant radiance being defined over infinitesimal solid angles 5Q and BRF can be calculated as an integrated property i.e. the numerator in equation 2.2 becomes:

2

L, (O, O’, E, (O, O' ), (O, O’ )) = I (O, O’ )dOdO’ 2.4

0 0

where ps is the surface reflectance function (BRDF); dCï= 6 sin OdOde/) (and similarly for dQ*. It can be seen from equation 2.3 that in order to derive the exitant radiance, the viewing and illumination vectors are integrated over the respective viewing

O ’.

Q '))= -

In

2.5 ^ 0

and illumination hemispheres. In the same way, the radiance from the Lambertian surface is defined as

The integral is purely over the illumination direction as the observed reflectance, by definition, is the same regardless o f the viewing direction. In practice, the BRF is often defined as the radiation exiting the scene in a given direction (an infinitesimal angle), rather than as an integrated property. If a point illumination source is considered i.e. no sky irradiance, then, for this case

B R F (£2, n ’ ) = n B R D F ( Q , Q ’ ) 2.6

The expression in equation 2.6 utilises the fact that the BRDF o f perfect Lambertian reflector is 1/ti. This is due to the fact that a perfectly diffuse surface reflects the same radiance, E/(0,)/7i in all directions. As a result, the BRF o f any surface is equal to its BRDF multiplied by tt.

Consequently, integrals o f the BRDF over the viewing and illumination hemispheres are defined. The directional hemispherical reflectance (or DHR), p ( Q ,Q ') , is the integral of BRF over the viewing (or illumination) hemisphere. It is the hemispherical reflectance assuming a directional (collimated beam) illumination source (or alternatively, the directional refleetanee for a diffuse illumination). This can be expressed as:

^ ( Q ’; 2;r) = - T BRDF{C1, Q')dCl 7T •’

This expression defines the so-called black-sky albedo (W anner et a l , 1995) i.e. it defines the hemispherical reflectance under conditions o f purely directional illumination (no sky radiance). yô(Q') can be integrated over all illumination directions to yield p the bi- hemispherical (or hemispherical-to-hemispherical) reflectance (or BHR). This is the reflectance o f a surface over all viewing angles due to a diffuse illumination source i.e.

27t 1 2;r In

p (2 ;r;2 ;r)= jp (Q ')rfQ ' = — I \B R D F {Q ,Q ')d Q d C l' 2.8 which is the so-called white-sky albedo i.e. the hemispherical reflectance under perfectly diffuse illumination conditions.

It is important in the context o f remote sensing applications to note that BRDF cannot be directly measured because:

i) (from equations 2.2 and 2.3) BRDF is defined as the ratio o f two partial derivatives and BRF is defined as reflectance relative to that o f a perfect Lambertian reflector.

ii) No sensor has a perfectly discrete spectral response, and so measurement o f BRDF is inevitably a convolution o f the signal with the sensor spectral response function over a range o f wavelengths. Additionally, the projected instantaneous IFOV o f the sensor must be accounted for i.e. measurement is not over an infinitesimal angle.

iii) In practice, observations are made through some depth o f atmosphere so that the measured signal is also a function o f atmospheric absorption and scattering (reflectance and transmittance). To retrieve measures o f surface reflectance through the atmosphere, the scattering behaviour of the

atmosphere (and in particular the scattering phase function) must be characterised accurately.

Using the relationships in equations 2.2-2.S it is now possible to relate BRDF to albedo. The BRDF explicitly describes the directional nature o f exitant radiation from a surface. As a result, albedo, the total irradiant energy (both direct and diffuse) reflected in all directions from the surface, is an integrated measure o f the directional reflectance over the viewing/illumination hemisphere. Albedo is thus a function o f the quantities o f diffuse and direct illumination arriving at the surface. In the discussion above, dependence o f BRDF on X has not been considered. In practice, BRDF is also a function o f Typically reflectance data are either reported as a function o f X, or the finite wavelength interval over which the parameters have been integrated during observation is specified. In such instances, parameters such as BRF, DHR and BHR are should be prefaces by "spectral" or "narrowband".

Further approximations are required in practice in order to relate estimates of albedo made at narrow wavelength bands (narrowband) to total albedo over the visible and NIR regions o f the spectrum (broadband). This has been discussed by a number of researchers, in a variety o f ways (Stephens et a l, 1981; Stum et a l, 1985; Kimes and Sellers, 1985; Cess and Potter, 1986; Brest and Goward, 1987; Koepke and Kriebel, 1987; Dickinson et a l, 1990; Liang et a l, 1999). Spectral albedo a ( l ) can be approximated as a combination o f the two components o f black-sky and white-sky albedo (Wanner et a l, 1997; Strugnell and Lucht, 1999), weighted by D, the proportion of diffuse illumination from the atmosphere i.e.

a( Z) = (l - D(à, Q ', t ))p {à ,Q ') + D{à, O ', t ) p { à ) 2.9

where D(^, Q ’, x) is function o f the illumination conditions and the atmospheric state, characterised by the atmospheric optical depth, x. Equation 2.9 is a reasonable approximation to spectral albedo except at high solar zenith angles (Lewis and Barnsley, 1994). The total surface albedo, a i.e. the ratio o f the total incident shortwave radiation to the reflected radiation can then be approximated as an integral over all shortwave (SW) radiation (ignoring anisotropy o f incident diffuse radiation) i.e.

a = jp{X)a{À)dÀ 2.10

where p (l) is the proportion o f illumination in the solar spectrum, dependent on the atmospheric state. 1- a is then the proportion o f incident shortwave radiation absorbed at the surface (stored as heat energy and chemical energy through photosynthesis). Application o f the method to EO data requires the integral over all SW to be approximated with a weighted sum o f the available wavebands (Lewis et a l, 1999), or through the use o f other modelling techniques (Liang, 2000; Lucht and Roujean, 2000). This limitation will be discussed in later chapters.

The properties defined above can be measured in practice. To characterise up- and downwelling radiation fluxes from the surface a description o f the surface scattering behaviour is required. Indeed, surface properties cannot be related to measured radiance without such a model. Many models describing the scattering o f radiation by vegetation have been developed; some to correct angular dependencies (Roujean et a l, 1992; Cihlar

et a l, 1994 Liang et a l, 2000b), some to relate canopy scattering behaviour to remote

measurements o f reflectance (e.g. Goel and Strebel, 1983; Goel 1988, 1992; Nilson and Kuusk, 1989; Pinty and Verstraete, 1991). An overview o f these methods is given in section 2.5.