# Radiometry and Reflectance: from Terminology Concepts to Measured

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Radiometry and Reﬂectance: From Terminology Concepts to Measured Quantities

Gabriela Schaepman-Strub, Michael Schaepman, John Martonchik, Thomas Painter, and Stefan Dangel

Keywords: radiometry, reﬂectance, terminology, reﬂectance measurements, reﬂectance quantities, albedo, reﬂectance factor, bidirectional reﬂectance, BRDF.

INTRODUCTION These differences are especially important in long term, large area trend studies, as the latter are The remote sensing community devotes major mostly based on multiple sensors with different efforts to calibrate sensors, improve measurement spectral and angular sampling, modeling, as well setups, and validate derived products to quan- as atmospheric correction schemes. tify and reduce measurement uncertainties. Given Optical remote sensing is based on the mea- recent advances in instrument design, radiomet- surement of reﬂected and emitted electromag- ric calibration, atmospheric correction, algorithm netic radiation. This chapter will deal with the and product development, validation, and delivery, reﬂected portion of optical remote sensing. Given the lack of standardization of reﬂectance termi- the inherent anisotropy of natural surfaces and nology and products has emerged as a source of the atmosphere, the observed reﬂected radiance considerable error. depends on the actual solar zenith angle, the Schaepman-Strub et al. (2006) highlighted the ratio of direct to diffuse irradiance (including fact that the current use of reﬂectance terminol- its angular distribution), the observational geom- ogy in scientiﬁc studies, applications, and pub- etry, including the swath width (ﬁeld of view, lications often does not comply with physical FOV) and the opening angle of the remote sens- standards. Biases introduced by using an inappro- ing instrument (i.e., the instantaneous ﬁeld of view, priate reﬂectance quantity can exceed minimum IFOV). Current atmospheric correction schemes sensitivity levels of climate models (i.e., ±0.02 compensate for the part of the observed signal reﬂectance units (Sellers et al. 1995)). Further, which is contributed by the atmosphere. How- they may introduce systematic, wavelength depen- ever, these schemes mostly rely on the assump- dent errors in reﬂectance and higher level prod- tion of Lambertian surfaces, thus neglecting their uct validation efforts, in data fusion approaches anisotropy and corresponding geometrical-optical based on different sensors, and in applications. effects introduced by the variation of illumination

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as well as differing IFOVs. Resulting at-surface Table 15.1 Symbols used in this chapter reﬂectance products may differ considerably by S distribution of direction of radiation physical deﬁnition further contributing to a numer- A surface area [m2] ical bias of products. 2 Ap projected area [m ] The aim of this chapter is to explain geometric- E irradiance, incident ﬂux density; − optical differences in remote sensing observations ≡ d/dA[Wm 2] − and reﬂectance quantities, in order to give users the I radiant intensity; ≡ d/dω[Wsr 1] − background to choose the appropriate product for L radiance; ≡ d2/(dA cos θdω)[Wm 2sr−1] their applications, to design experiments with ﬁeld M radiant exitance, exitent ﬂux density; − instrumentation, and process the measurements ≡ d/dA[Wm 2] accordingly. Q energy [J] This chapter presents a systematic and consistent ρ reﬂectance; ≡ d /d [dimensionless] deﬁnition of radiometric units, and a conceptual r i R reﬂectance factor; ≡ d /did [dimensionless] model for the description of reﬂectance quanti- r r t time ties. Reﬂectance terms such as BRDF, HDRF, β plane angle [rad] BRF, BHR, DHR, black-sky albedo, white-sky radiant ﬂux, power [W] albedo, and blue-sky albedo are deﬁned, explained, θ zenith angle, in a spherical coordinate system [rad] and exempliﬁed, while separating conceptual from φ azimuth angle, in a spherical coordinate measurable quantities. The reﬂectance conceptual system [rad] model is used to specify the measured quantities ω solid angle; ≡ dω ≡ sin θdθdφ[sr] of current laboratory, ﬁeld, airborne, and satel- projected solid angle; lite sensors. Finally, the derivation of higher-level ≡ cos θdω ≡ cos θ sin θdθdφ[sr] reﬂectance products is explained, followed by λ wavelength of the radiation [nm] examples of operational products. All symbols and main abbreviations used in this chapter are listed Sub- and superscripts in Table 15.1. iincident rreﬂected id ideal (lossless) and d iffuse (isotropic or Lambertian) RADIOMETRY AND atm atmospheric GEOMETRICAL-OPTICAL CONCEPTS dir dir ect diff diff use Radiometry is the measurement of optical radi- Terms ation, which is electromagnetic radiation within BHR BiHemispherical Reﬂectance the wavelength range 0.01–1000 micrometers BRDF Bidirectional Reﬂectance Distribution Function (µm). Photometry follows the same deﬁnition as BRF Bidirectional Reﬂectance Factor radiometry, except that the measured radiation is DHR Directional – Hemispherical Reﬂectance weighted by the spectral response of the human HDRF Hemispherical – Directional Reﬂectance Factor eye. Photometry is thus restricted to the wave- length range from about 360 to 830 nanometers (nm; 1000 nm = 1µm), typical units used in pho- tometry include lumen, lux, and candela. Remote • Power, also known as radiant ﬂux, is another SI sensing detectors are usually not adapted to the derived unit. It is the derivative of energy with response function of the human eye; therefore this respect to time, dQ/dt, and the unit is the watt (W). chapter concentrates on radiometry. The following The recommended symbol for power is . Energy two sections on radiometry are primarily based on is the integral over time of power, and is used for an extended discussion of Palmer (2003). integrating detectors and pulsed sources, whereas power is used for non-integrating detectors and continuous sources. Basic quantities and units - energy, power, projected area, solid angle (b) Geometry Radiometric units are based on two conceptual • The projected area, Ap , is deﬁned as the recti- approaches, namely those based on (a) power or linear projection of a surface of any shape onto energy, or (b) geometry. a plane normal to the unit vector (Figure 15.1, top). The differential form is dAp = cos(θ)dA (a) Power and energy where θ is the angle between the local surface • Energy is an SI derived unit, measured in Joules, normal and the line of sight. The integration over with the recommended symbol Q. the surface area leads to Ap = cos(θ)dA. A

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• The solid angle, ω, extends the concept of the plane angle to three dimensions, and equals A q the ratio of the spherical area to the square of the radius (Figure 15.1, bottom). As an example we Ap consider a sphere with a radius of 1 metre. A cone that covers an area of 1 m2 on the surface of the sphere encloses a solid angle of 1 steradian (sr). A full sphere has a solid angle of 4π steradian. A round object that appears under an angle of 57◦ subtends a solid angle of 1 sr. In comparison, the sun covers a solid angle of only 0.00006 sr when viewed from Earth, corresponding to a plane angle ◦ S = 1 of 0.5. The projected solid angle is deﬁned as = cos(θ)dω. 1 rad b Radiometric units - irradiance, radiance, r = 1 reﬂectance, reﬂectance factor, and wavelength dependence Referring to the above concepts, we can now b = s/r approach the basic radiometric units as used in w remote sensing (Figure 15.2). r = 1 • Irradiance, E (also know as incident ﬂux density), is measured in W m−2. Irradiance is power per unit area incident from all directions in a hemisphere onto a surface that coincides with the base of that A = 1 hemisphere (d/dA). A similar quantity is radiant exitance, M, which is power per unit area leaving 1sr a surface into a hemisphere whose base is that surface. • Reﬂectance, ρ, is the ratio of the radiant exitance (M [W m−2]) with the irradiance (E [W m−2]), and Figure 15.1 Top: Projected area A . Middle: p as such dimensionless. Following the law of energy Plane angle β. Bottom: Solid angle ω. conservation, the value of the reﬂectance is in the (Reproduced with permission from inclusive interval 0 to 1. International Light Technologies Inc, − • Radiant intensity , I, is measured in W sr 1, and Peabody, MA). is power per unit solid angle (d/dω). Note that the atmospheric radiation community mostly uses the terms intensity and ﬂux as they were deﬁned in Chandrasekhar’s classic work (Chandrasekhar • The plane angle is deﬁned as the length of 1950). More recent textbooks on atmospheric an arc (s) divided by its radius (r ), β = s/r radiation still propose the use of intensity, with (Figure 15.1, middle). If the arc that is subtended units W m−2 sr−1, along with a footnote say- by the angle is exactly as long as the radius of ing that intensity is equivalent to radiance. An the circle, then the angle spans 1 radian. This is extensive discussion on the (mis-)usage of the equivalent to about 57.2958◦. A full circle cov- term intensity and corresponding units is given in ers an angle of 2π radians or 360◦, therefore the Palmer (1993). He concludes that following the SI conversion between degrees and radians is 1 rad system deﬁnition of the base unit candela, the SI = (180/π) degrees. In SI-terminology the above derived unit for radiant intensity is W sr−1, and for reads as follows: The radian is the plane angle radiance W m−2 sr−1. In this chapter, we follow between two radii of a circle that cuts off on the the SI deﬁnitions. circumference an arc equal in length to the radius • Radiance, L, is expressed in the unit W m−2 sr−1, (Taylor 1995). and is power per unit projected area per unit solid

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Ω 2 Ap m W sr L m2sr E = LΩ L = IAp ÷ X

W W I E sr m2

X ÷ Φ = Iω E = Φ/A F W ω A sr m2

Figure 15.2 Overview of radiometric quantities (left) and corresponding units (right) as used in remote sensing, namely radiant intensity I, radiance L, irradiance E, and radiant ﬂux (or power) . (See the color plate section of this volume for a color version of this ﬁgure.)

angle (d2/dω dA cos(θ), where θ is the angle Lambertian always refers to a ﬂat surface with between the surface normal and the speciﬁed the reﬂected intensity falling off as the cosine direction). of the observation angle with respect to the sur- • The reﬂectance factor, R, is the ratio of the radi- face normal (Lambert’s cosine law). Isotropic on ant ﬂux reﬂected by a surface to that reﬂected the other hand means ‘having the same prop- into the same reﬂected-beam geometry and wave- erties in all directions’, and does not refer to a length range by an ideal (lossless) and diffuse speciﬁc physical quantity. Therefore, a perfectly (Lambertian) standard surface, irradiated under diffuse or Lambertian surface element dA is one the same conditions. Reﬂectance factors can reach for which the reﬂected radiance is isotropic, with values beyond 1, especially for strongly forward the same value for all directions into the full hemi- reﬂecting surfaces such as snow (Painter and sphere above the element dA of the reﬂecting Dozier 2004). For measurement purposes, a Spec- surface. tralon panel commonly approximates the ideal diffuse standard surface. This is a manufactured standard having a high and stable reﬂectance Geometrical-optical concepts – throughout the optical region, approximates a directional, conical, and hemispherical Lambertian surface, and is traceable to the U.S. National Institute of Standards and Technology The anisotropic reﬂectance properties of a sur- (NIST). Its use in ﬁeld spectroscopy and additional face (Figure 15.3) can mathematically be described by the bidirectional reﬂectance distribution func- references are described in Milton et al. (in press). tion (BRDF). The term bidirectional implies single We assume further that an isotropic behavior directions for the incident and reﬂected radiances implies a spherical source that radiates the same (entering and emanating from differential solid −1 in all directions, i.e., the intensity [W sr ]isthe angles, respectively). This mathematical concept same in all directions. The Lambertian behavior can only be approximated by measurements, since refers to a ﬂat reﬂective surface. The intensity inﬁnitesimal elements of solid angle do not include of light reﬂected from a Lambertian surface falls measurable amounts of radiant ﬂux (Nicodemus off as the cosine of the observation angle with et al. 1977), and unlimited small light sources, as respect to the surface normal (Lambert’s cosine well as an unlimited small sensor instantaneous law), whereas the radiance L [W m−2 sr−1]is ﬁeld of view (IFOV) do not exist. Consequently, all measurable quantities of reﬂectance are per- independent of observation angle. The reﬂected formed either in the conical or hemispherical radiant ﬂux from a given area is reduced by domain. From a physical point of view, we there- the cosine of the observation angle, but the fore differentiate between conceptual (directional) observed area has increased by the cosine of the and measurable quantities (involving conical and angle, and therefore the observed radiance is the hemispherical solid angles of observation and illu- same independent of observation angle. Note that mination). According to Nicodemus et al. (1977),

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Distribution Function), for example when measuring and modeling leaf optical properties. We symbolize reﬂectance and reﬂectance factor as

ρ(Si, Sr,λ) = reﬂectance, and R (Si, Sr,λ) = reﬂectance factor

where Si and Sr describe the angular distribution of all incoming and reﬂected radiation observed by the sensor, respectively. Si and Sr only describe a set of angles occurring with the incoming and reﬂected radiation and not their intensity distribu- Figure 15.3 Reﬂectance anisotropy of a tions. Sr represents a cone with a given solid angle vegetation canopy showing the dependence corresponding to a sensor’s instantaneous ﬁeld of view (IFOV), but no sensor weight functions are of the observed reﬂectance on the viewing included here. This becomes only necessary if the direction and IFOV of the remote sensing sensitivity of the sensor depends on the location instrument (black). The shape of the within the rim of the cone. When a sensor has a dif- reﬂectance distribution changes with solar ferent IFOV for different wavelength ranges, then angle and ratio of direct solar radiation and Sr depends on the wavelength. diffuse (radiation scattered by the The terms Si and Sr can be expanded into a more atmosphere) illumination. It can also be seen explicit angular notation to address the remote that vegetation is mainly a backward sensing problem: scattering object, with a so-called ρ θ ,φ,ω; θ ,φ ,ω ; λ , reﬂectance hot spot toward the main ( i i i r r r ) and illumination direction (as opposed to snow, R (θi,φi,ωi; θr,φr,ωr; λ) which is a forward scatterer), while waxy θ φ leaves may introduce a forward scattering where the directions ( and are the zenith and component. azimuth angle, respectively) of the incoming (sub- script i) and the reﬂected (subscript r) radiation, and the associated solid angles of the cones (ω) are the angular characteristics of the incoming radiance indicated. This notation follows the deﬁnition of a are named ﬁrst in the reﬂectance term, followed by general cone. the angular characteristics of the reﬂected radiance. For surface radiation measurements made from This results in nine different cases of reﬂectance space, aircraft or on the ground, under ambient sky quantities, illustrated in Figure 15.4. The math- conditions, the cone of the incident radiation is ω = π[ ] ematical derivations for the different cases are of hemispherical extent ( 2 sr ). The inci- given below, followed by sections elaborating the dent radiation may be divided into a direct sunlight geometrical conﬁgurations of most common mea- component and a second component, namely sun- surement setups and on the resulting recommended light which has been scattered by the atmosphere, terminology. the terrain, and surrounding objects, resulting in an anisotropic, diffuse illumination, sometimes called ‘skylight’. The above reﬂectance and reﬂectance factor REFLECTANCE QUANTITIES IN REMOTE deﬁnitions lead to the following special cases: SENSING – BRDF, BRF, HDRF, DHR, BHR • ωi or ωr are omitted when either is zero (direc- Based on the above concepts, we can develop tional quantities). • the corresponding mathematical formula- If 0 < (ωi or ωr ) < 2π, then θ, φ describe the tions for the most relevant quantities used direction of the center axis of the cone (e.g., the in remote sensing, namely the BRDF (Bidi- line from a sensor to the center of its ground ﬁeld rectional Reﬂectance Distribution Function), of view – conical quantities). BRF (Bidirectional Reﬂectance Factor), HDRF • ω = π θ ,φ (Hemispherical-Directional Reﬂectance Factor), If i 2 , the angles i i indicate the direc- DHR (Directional-Hemispherical Reﬂectance), tion of the incoming direct radiation (e.g., the posi- and BHR (Bihemispherical Reﬂectance). The same tion of the sun). For remote sensing applications, concepts may be extended to the transmittance it is often useful to separate the natural incoming behavior (BTDF, Bidirectional Transmittance radiation into a direct (neglecting the sun’s size)

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Incoming/Reflected Directional Conical Hemispherical Directional Bidirectional Directional-conical Directional-hemispherical Case 1 Case 2 Case 3

Conical Conical-directional Biconical Conical-hemispherical Case 4 Case 5 Case 6

Hemispherical Hemispherical-directional Hemispherical-conical Bihemispherical Case 7 Case 8 Case 9

Figure 15.4 Geometrical-optical concepts for the terminology of at-surface reﬂectance quantities. All quantities including a directional component (i.e., Cases 1–4, 7) are conceptual quantities, whereas measurable quantities (Cases 5, 6, 8, 9) are shaded in gray. (Reprinted from G., Schaepman-Strub, M. E. Schaepman, T. H. Painter, S. Dangel, and J. V. Martonchik, 2006. Reﬂectance quantities in optical remote sensing–deﬁnitions and case studies. Remote Sensing of Environment, 103: 27–42). (See the color plate section of this volume for a color version of this ﬁgure.)

and hemispherical diffuse part. One may also Based on this implication, Martonchik et al., 2000, include a terrain reﬂected diffuse component that adapted the terminology to the remote sensing case, is calculated with a topographic radiation model which involves direct and diffuse sky illumina- such as TOPORAD (Dozier 1980). Consequently, tion. In the following, we give the mathematical the preferred notation for the geometry of the description of the most commonly used quantities in remote sensing, thus the general expressions for incoming radiation under ambient illumination θ ,φ , π non-isotropic incident radiation. When applicable, conditions is i i 2 . Note that in this case, we simplify the expression for the special case of θ ,φ i i describe the position of the sun and not the isotropic incident radiation. Further, the particular center of the cone (2π). In the case of an isotropic wavelength dependency is omitted as well in most diffuse irradiance ﬁeld, without any direct irradi- cases to improve readability of the equations. How- ance component (closest approximated in the case ever, it must be understood that all interaction of of an optically thick cloud deck), θi ,φi are omit- light with matter is wavelength dependent, and may ted. Isotropic behavior implies that the intensity not simply be ignored. [W sr−1] is the same in all directions. • If ωr = 2π, θr and φr are omitted. The bidirectional reﬂectance It should be noted that the nine standard distribution function (BRDF) – Case 1 reﬂectance terms deﬁned by (Nicodemus et al. 1977) ‘are applicable only to situations with uni- The bidirectional reﬂectance distribution function form and isotropic radiation throughout the inci- (BRDF) describes the scattering of a parallel beam dent beam of radiation’. They then state that, of incident light from one direction in the hemi- ‘If this is not true, then one must refer to the sphere into another direction in the hemisphere. more general expressions’. This implies that any The term BRDF was ﬁrst used in the literature in the signiﬁcant change to the nine reﬂectance concepts early 1960s (Nicodemus 1965). Being expressed when the incident radiance is anisotropic lies in the as the ratio of inﬁnitesimal quantities, it cannot be mathematical expression used in their deﬁnition. directly measured (Nicodemus et al. 1977). The

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BRDF describes the intrinsic reﬂectance proper- the deﬁnition of the BRF, but includes irradiance ties of a surface and thus facilitates the derivation from the entire hemisphere. This makes the quan- of many other relevant quantities, e.g., conical tity dependent on the actual, simulated or assumed and hemispherical quantities, by integration over atmospheric conditions and the reﬂectance of the corresponding ﬁnite solid angles. surrounding terrain. This includes spectral effects The spectral BRDF, fr(θi,φi; θr,φr; λ) can be introduced by the variation of the diffuse to direct expressed as: irradiance ratio with wavelength (e.g., Strub et al. 2003). BRDFλ = fr (θi,φi; θr ,φr ; λ)

dL (θ ,φ; θ ,φ ; λ) − HDRF = r i i r r [sr 1]. (1) dEi(θi,φi; λ) d (θ ,φ, 2π ; θ ,φ ) = R(θ ,φ, 2π; θ ,φ ) = r i i r r (7) i i r r id θ ,φ, π d r ( i i 2 ) For reasons of clarity, we will omit the spectral cos θ sin θ L (θ ,φ, 2π ; θ ,φ )dθ dφ dA dependence in the following. We therefore write = r r r i i r r r r (8) θ θ id θ ,φ, π θ φ for the BRDF: cos r sin r Lr ( i i 2 )d r d r dA Lr (θi,φi, 2π ; θr ,φr ) = θ ,φ; θ ,φ = BRDF fr ( i i r r ) Lid (θ ,φ, 2π) r i i L θ ,φ; θ ,φ = d r ( i i r r ) [ −1]. f (θ ,φ; θ ,φ )d (θ ,φ) θ ,φ sr (2) = 2π r i i r r i i i dEi( i i) /π θ ,φ (9) 2π (1 )d i( i i) π π/ 2 2 f (θ ,φ; θ ,φ ) cos θ sin θ L (θ ,φ)dθ dφ = 0 0 r i i r r i i i i i i i . π π/ (1/π) 2 2 cos θ sin θ L (θ ,φ)dθ dφ Reﬂectance factors – Deﬁnition of 0 0 i i i i i i i Cases 1, 5, 7 and 8 (10) When reﬂectance properties of a surface are mea- If we divide Li into a direct (Edir with angles sured, the procedure usually follows the deﬁnition θ0,φ0) and diffuse part, we may continue: of a reﬂectance factor. The reﬂectance factor is the ⎛ ⎞ ratio of the radiant ﬂux reﬂected by a sample sur- fr (θ0,φ0;θr ,φr )Edir (θ0,φ0) ⎜ ⎟ face to the radiant ﬂux reﬂected into the identical ⎝ 2π π/ 2 ⎠ + θ ,φ ;θ ,φ θ θ diff θ ,φ θ φ beam geometry by an ideal (lossless) and diffuse fr ( i i r r )cos i sin iLi ( i i)d id i (Lambertian) standard surface, irradiated under the = 0⎛0 ⎞ (1/π)(Edir (θ0,φ0) same conditions as the sample surface. ⎜ ⎟ ⎝ 2π π/ 2 ⎠ The bidirectional reﬂectance factor (BRF; + θ θ diff θ ,φ θ φ cos i sin iLi ( i i)d id i) Case 1) is given by the ratio of the reﬂected radiant 0 0 ﬂux from the surface area dA to the reﬂected radiant (11) ﬂux from an ideal and diffuse surface of the same area dA under identical view geometry and single diff direction illumination: then, if and only if Li is isotropic (i.e., indepen- dent of the angles), we may continue: dr (θi,φi; θr ,φr ) ⎛ ⎞ BRF = R(θi,φi; θr ,φr ) = (3) id θ ,φ fr (θ0,φ0; θr ,φr )Edir (θ0,φ0) d r ( i i) ⎜ ⎟ ⎝ 2π π/ 2 ⎠ diff cos θr sin θr dLr (θi,φi; θr ,φr )dθr dφr dA +L f (θ ,φ; θ ,φ ) cos θ sin θ dθ dφ = (4) i r i i r r i i i i θ θ id θ ,φ θ φ 0 0 cos r sin r dLr ( i i)d r d r dA = 2π π/ 2 dE (θ ,φ) dL (θ ,φ; θ ,φ ) (1/π) E (θ ,φ ) + Ldiff cos θ sin θ dθ dφ = i i i · r i i r r (5) dir 0 0 i i i i i id θ ,φ θ ,φ 0 0 dLr ( i i) dEi( i i) (12) fr (θi,φi,θr ,φr ) = = πfr (θi,φi; θr ,φr ). (6) /π θ ,φ f id (θ ,φ) (1 )Edir ( 0 0) r i i = π fr (θ0,φ0; θr ,φr ) /π θ ,φ + diff (1 )Edir ( 0 0) Li An ideal Lambertian surface reﬂects the same 2π π/2 radiance in all view directions, and its BRDF + fr (θi,φi; θr ,φr ) cos θi sin θidθidφi is 1/π. Thus, the BRF [unitless] of any surface 0 0 − can be expressed as its BRDF [sr 1] times π Ldiff × i (13) (Equation (6)). For id and Lid, we omit the diff r r (1/π)Edir (θ0,φ0) + L view zenith and azimuth angles, because there is i no angular dependence for the ideal Lambertian = R(θ0,φ0; θr ,φr )d + R(2π ; θr ,φr )(1 − d) (14) surface. The concept of the hemispherical-directional where d corresponds to the fractional amount of reﬂectance factor (HDRF; Case 7) is similar to direct radiant ﬂux (i.e., d∈[0, 1]).

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The biconical reﬂectance factor (conical– with a parallel beam of light from a single conical reﬂectance factor, CCRF; Case 5), is direction. deﬁned as: DHR CCRF θ ,φ ; π = ρ θ ,φ ; π = d r ( i i 2 ) = θ ,φ ,ω ; θ ,φ ,ω ( i i 2 ) R( i i i r r r ) di(θi,φi) 2π π/2 f (θ ,φ ; θ ,φ )L (θ ,φ )d d dA dLr (θi,φi; θr ,φr ) cos θr sin θr dθr dφr r i i r r i i i i r = 0 0 (17) ωr ωi θ ,φ = (15) d i( i i) π θ ,φ r Li( i i)d i π π/2 ω d (θ ,φ ) 2 f (θ ,φ ; θ ,φ ) cos θ sin θ dθ dφ i = i i i 0 0 r i i r r r r r r θ ,φ d i( i i) = = θ ω = θ θ (18) where d cos d cos sin dθdφ is the projected solid angle of the cone. 2π π/2 = fr (θi,φi; θr ,φr ) cos θr sin θr dθr dφr . (19) Formally, the CCRF can be seen as the most gen- 0 0 eral quantity, because its expression contains all other cases as special ones: for ω = 0 the inte- The bihemispherical reﬂectance (BHR; Case 9), gral collapses and we obtain the directional case, generally called albedo, is the ratio of the radi- and for ω = 2π we obtain the hemispherical case. ant ﬂux reﬂected from a unit surface area into the However, the BRF and BRDF remain the most fun- whole hemisphere to the incident radiant ﬂux of damental and desired quantities because they are hemispherical angular extent: the only quantities not integrated over a range of angles. BHR

For large IFOV measurements performed under dr (θi,φi, 2π ; 2π) = ρ(θi,φi, 2π ; 2π) = (20) ambient sky illumination, the assumption of a zero di(θi,φi, 2π) interval of the solid angle for the measured reﬂected 2π π/2 dA dLr (θi,φi, 2π ; θr ,φr ) cos θr sin θr dθr dφr radiance beam does not hold true. The result- = 0 0 2π π/2 θ ,φ θ θ θ φ ing quantity most precisely could be described as dA 0 0 dLi( i i) cos i sin id id i (21) hemispherical-conical reﬂectance factor (HCRF; ⎛ ⎞ Case 8), obtained from Equation (15) by setting 2π π/ 2 2π π/ 2 ⎜ θ ,φ ; θ ,φ θ θ θ φ ⎟ ω = 2π: ⎝ fr ( i i r r ) cos r sin r d r d r ⎠ i 0 0 0 0 Li(θi,φi) cos θi sin θidθidφi HCRF = R(θ ,φ , 2π ; θ ,φ ,ω ) = i i r r r 2π π/ 2 L (θ ,φ ) cos θ sin θ dθ dφ θ ,φ ; θ ,φ θ ,φ i i i i i i i fr ( i i r r )Li( i i)d id r 0 0 ωr π = 2 . (16) (22) r π Li(θi,φi)di π 2π π/2 2 ρ(θi,φi; 2π)Li(θi,φi) cos θi sin θidθidφi = 0 0 . (23) 2π π/2 θ ,φ θ θ θ φ 0 0 Li( i i) cos i sin id id i Reﬂectance – Deﬁnition of Cases 3 and 9 If as before we divide Li into a direct (Edir with θ ,φ When applying remote sensing observations to angles 0 0) and diffuse part, and assume that diff surface energy budget studies, for example, the Li is isotropic we can write: total energy reﬂected from a surface is of inter- ⎛ ⎞ ρ(θ ,φ ; 2π)E (θ ,φ ) est, rather than a reﬂectance quantity directed ⎜ 0 0 dir 0 0 ⎟ ⎜ ⎟ ⎝ 2π π/ 2 ⎠ into a small solid angle. In the following, we +π diff /π ρ θ ,φ ; π θ θ θ φ describe the hemispherical reﬂectance as a func- Li (1 ) ( i i 2 ) cos i sin id id i = 0 0 tion of different irradiance scenarios including θ ,φ + π diff Edir ( 0 0) Li (i) the special condition of pure direct irradiance, (24) (ii) common Earth irradiance, composed of dif- = ρ θ ,φ ; π + ρ π; π − fuse and direct components, and (iii) pure diffuse ( 0 0 2 )d (2 2 )(1 d) (25) irradiance. The directional-hemispherical reﬂectance where d again corresponds to the fractional amount (DHR; Case 3) corresponds to pure direct illumi- of direct radiant ﬂux. nation (reported as black-sky albedo in the MODIS For the special case of pure diffuse isotropic inci- (Moderate Resolution Imaging Spectroradiometer) dent radiation, a situation that may be most closely product suite (Lucht et al. 2000)). It is the ratio of approximated in the ﬁeld by a thick cloud or aerosol the radiant ﬂux for light reﬂected by a unit surface layer, the resulting BHR (reported as white-sky area into the view hemisphere to the illumination albedo in the MODIS product suite (Lucht et al. radiant ﬂux, when the surface is illuminated 2000), and sometimes also referred to as BHRiso)

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can be described as follows: non-parallel (due to internal beam divergence and collimating limitations), whereas solar direct illu- BHR=ρ(2π;2π) mination can be approximated as being parallel (within 0.5◦). The diffuse illumination compo- 1 2π π/2 = ρ(θi,φi;2π)cosθi sinθidθidφi. nent in the laboratory can be minimal when the π 0 0 reﬂections are minimized (e.g., walls are painted (26) black and black textiles cover reﬂecting objects). For the Laboratory Goniometer System (LAGOS), Under ambient illumination conditions, the the diffuse-to-total illumination ratio was shown albedo is inﬂuenced by the combined diffuse and to be lower than 0.5% in the spectral range of direct irradiance. To obtain an approximation of the 400–1000 nm (Dangel et al. 2005). The instanta- albedo for ambient illumination conditions (also neous ﬁeld of view (IFOV) of a few degrees of the reported as blue-sky albedo in the MODIS product non-imaging spectroradiometer employed corre- suite), it is suggested that the BHR for isotropic sponds to a conical opening angle. Given the above diffuse illumination conditions and the DHR be conditions, the typical measurement setup of labo- combined linearly (see Equation (25)), correspond- ratory spectrometer measurements corresponds to ing to the actual ratio of diffuse to direct illumina- the biconical conﬁguration, resulting in conical– tion (Lewis and Barnsley 1994, Lucht et al. 2000). conical reﬂectance factors (CCRF – Case 5). For a The diffuse component then can be expressed as a perfectly collimated light source and a small IFOV, function of wavelength, optical depth, aerosol type, measurements may approximate the bidirectional and terrain contribution. The underlying assump- quantity (e.g., the SpectroPhotoGoniometer (SPG) tion of an isotropic diffuse illumination may lead to to measure leaf optical properties (Combes et al. signiﬁcant uncertainties due to ignoring the actual 2007)). distribution of the incoming diffuse radiation (e.g., Pinty et al. 2005). All the above-mentioned albedo values, with the Ground based ﬁeld instruments exception of the BHR for pure diffuse illumination In the ﬁeld, ambient illumination always includes a conditions, depend on the actual illumination angle diffuse fraction. Its magnitude and angular distribu- of the direct component. Thus it is highly recom- tion depend on the actual atmospheric conditions, mended to include the illumination geometry in the surrounding terrain and objects, and wavelength. metadata of albedo quantities. Thus, outdoor measurements always include hemi- spherical illumination, which can be described as a composition of a direct and an anisotropic OBSERVATIONAL GEOMETRY OF REMOTE diffuse component. Shading experiments are dis- SENSING INSTRUMENTS cussed in Schaepman-Strub et al. (2006), where it is concluded that they are only suitable to sepa- This section discusses the geometric conﬁguration rate direct and diffuse illumination if the shading of selected operational sensors, including labo- object exactly covers the solar disc (0.00006 sr). ratory and ﬁeld instruments, as well as airborne The reason is that a signiﬁcant fraction of diffuse and spaceborne sensors, using the basic concep- illumination is located within a small cone in the tual model as presented in Figure 15.4. From direction of the direct illumination of the sun. a strict physical point of view, the most com- The partitioning into direct and diffuse illu- mon measurement setup of satellites, airborne, and mination inﬂuences the radiation regime within ﬁeld instruments corresponds to the hemispherical- vegetation canopies. Based on a modeling study, conical conﬁguration (Case 8), while laboratory Alton (2007) showed that the light use efﬁ- conditions are mostly biconical (Case 5). ciency (LUE) of three forest canopies increases by 6–33% when the irradiance is dominated by dif- fuse rather than direct sunlight. This demonstrates Laboratory instruments the importance of accompanying ﬁeld spectrometer campaigns with sun photometer measurements to Laboratory conditions provide the ability to assess the contribution of direct and diffuse irra- measure reﬂectance properties under controlled diance. For ﬁeld instruments with an IFOV full environmental conditions, being independent of cone angle of about 4–5 degrees (e.g., PARABOLA irradiance variations due to a changing atmosphere, (Portable Apparatus for Rapid Acquisition of Bidi- time of the day, or season. This is desirable when rectional Observation of the Land andAtmosphere, inherent reﬂectance properties (i.e., the BRDF) of a Abdou et al. 2001) or ASG (Automated Spectro- surface are investigated. Laboratory measurements Goniometer, Painter et al. 2003)), the surface direc- involve an artiﬁcial light source, which is usually tional reﬂectance variability across the opening

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angle needs to be investigated. As long as this Multi-angular sampling principles variability is unknown or not neglectable and cor- rected for, the measurements should be reported All approaches correcting for the bias introduced as HCRF (Case 8). This is especially true for by varying sun and view angles rely on multi- sensors with larger IFOV, such as the ASD (Ana- angular information to infer the BRDF as intrinsic lytical Spectral Devices) FieldSpec series (25◦, reﬂectance property of the surface. The BRDF is a while fore-optics allow a restriction to 8◦ or less), function of the solar and observational angles, thus and in cases where the sensitivity of the sen- measurements are performed under changing illu- sor outside of the cone only gradually falls off mination or viewing geometries or a combination across several degrees outside of the half power of both. Instantaneous multi-angular sampling is point. More details concerning ﬁeld spectrome- very rare, as most sampling schemes rely on tilting ter measurements can be found in Milton et al. sensors and thus changing their viewing geometry. (in press). The MISR satellite has nine cameras with ﬁxed Albedometers are designed to cover the full viewing angles, and approximately 7 min lapses down- and upward hemisphere (two pyranometers between the ﬁrst and the last camera overpass for a with an IFOV of 180◦ each) and approximate the selected area. For non-instantaneous multi-angular bihemispherical conﬁguration (Case 9) (e.g., Kipp measuring concepts, assumptions on the temporal and Zonen 2000). stability of the surface or the atmospheric compo- sition (e.g., aerosol optical depth) are often made. This is a disadvantage for highly variable surfaces such as vegetation canopies which change their Airborne sensors physiological state throughout a day, or snowmelt The surface illumination conditions for airborne events lasting several days. A selection of the most sensor observations are the same as for ﬁeld common multi-angular sampling principles of lab- measurements, thus of hemispherical extent (see oratory, ﬁeld, airborne, and satellite sensors is given above). The IFOV of airborne sensors is usually in Figure 15.5. very small, e.g., 0.021◦ for the Airborne Multi- angle Imaging SpectroRadiometer (AirMISR), 0.057◦ for the Airborne Visible/InfraRed Imag- ing Spectrometer (AVIRIS), 0.189◦ for the Dig- PROCESSING OF REFLECTANCE ital Airborne Imaging Spectrometer 7915 (DAIS PRODUCTS 7915), and 0.129◦ for the HyMap airborne hyperspectral scanner. In a strict physical sense, While the preceding section explained the observa- airborne observations therefore correspond to tion geometry of operational sensors and the multi- the hemispherical-conical conﬁguration (HCRF – angular sampling principles, this section will focus Case 8), while numerically approaching the on the derivation of various reﬂectance quantities hemispherical-directional conﬁguration (HDRF – from the observations. Case 7). Most correction schemes for airborne data Most state of the art atmospheric correction do not correct for the hemispherical irradiance and schemes convert top of atmosphere radiance to one thus the resulting at-surface reﬂectances approx- singular view angle at-surface reﬂectance, while imately correspond to HDRFs (Schaepman-Strub preserving the inﬂuence of the diffuse illumina- et al. 2006). tion on the surface reﬂectance, thus representing HDRF data. However, this reﬂectance quantity does not exactly represent what is required for Satellite sensors many applications, such as energy budget studies, multi-temporal investigations, and studies rely- Ambient illumination conditions of hemispheri- ing on multiple sensor data. The main product cal extent are also present at the Earth surface pathways to obtain higher level reﬂectance prod- when observed from spaceborne sensors. Gen- ucts are discussed below, namely (a) removing erally, space-based instruments with a spatial the effect of the diffuse hemispherical illumina- resolution of about 1 km have an IFOV with tion in single view angle observations to obtain a full cone angle of approximately 0.1◦ (e.g., inherent reﬂectance properties of the surface (i.e., Multiangle Imaging SpectroRadiometer (MISR), the BRDF), (b) interpolating and extrapolating the MODerate resolution Imaging Spectroradiome- single-angle observations to the entire reﬂected ter (MODIS), Advanced Very High Resolution hemisphere to obtain albedo quantities, and (c) Radiometer (AVHRR)). If the HDRF is constant normalizing the single-angle observations to a over the full cone angle of the instrument IFOV, standardized viewing geometry (i.e., to compute then the HCRF numerically equals the HDRF. This Nadir BRDF Adjusted Reﬂectance, NBAR). All approximation is mostly used when processing three approaches are based on the derivation of satellite sensor data. the BRDF, thus on the extraction of the intrinsic

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Figure 15.5 Examples of multi-angular sampling principles: (a) laboratory facility to measure leaf optical properties (SPG) (photo courtesy of Stéphane Jacquemoud), (b) ﬁeld goniometer system (FIGOS), (c) airborne multi-angular sampling during DAISEX’99 using the HyMap sensor (SZ = solar zenith angle (Berger et al. 2001)), (c) spaceborne near-instantaneous multi-angular sampling (MISR with nine cameras), (e) daily composites of geostationary satellite sensors (e.g., Meteosat), (f) multiple-day compositing of polar orbiting satellite sensors (e.g., MODIS 16 days). (See the color plate section of this volume for a color version of this ﬁgure.)

reﬂectance properties of the surface, using multi- Currently, most of the existing processing angular sampling of the observations through vari- approaches assume that the HDRF is constant ation of sun and/or viewing angles. The derivation over the full cone angle of the instrument of the BRDF based on laboratory measurements IFOV, thus the HCRF numerically equals the requires a correction for conicity and inhomo- HDRF (Case 7) without further correction. Given geneity of the artiﬁcial illumination source (for a sufﬁcient number of viewing angles (e.g., details see Dangel et al. 2005). The derivation of different at-surface reﬂectance quantities from MISR), the BHR is directly derived through measurements usually requires a sophisticated pro- interpolation. The algorithm for retrieving the cessing scheme (Figure 15.6), as for example HDRF and BHR from MISR top-of-atmosphere implemented for the MISR surface reﬂectance (TOA) radiances is virtually independent of any products (Martonchik et al. 1998). Unfortunately, particular kind of surface BRF model and its this issue does not always receive sufﬁcient atten- accuracy mainly depends on the accuracy of the tion in remote sensing when implementing process- atmospheric information used (Martonchik et al. ing schemes for ‘reﬂectance’ products. Below, the 1998). main required processing steps to infer the whole 2 HDRF (Case 7) → BRDF and BRF (Case 1) suite of reﬂectance quantities are described, includ- BRF data are derived using a parameterized ing some examples as implemented in operational algorithms, and the assumptions used. For each BRDF model to eliminate the diffuse illumina- heading, the input reﬂectance quantity of the pro- tion effects present in the HDRFs (e.g., Modi- cessing scheme is speciﬁed on the left hand side of ﬁed Rahman Pinty Verstraete (MRPV) for MISR the arrow, and the resulting reﬂectance quantity on (Martonchik et al. 1998) and ground based mea- the right. surements (Lyapustin and Privette 1999)). An alternative approach is used for MODIS, where 1 HCRF (Case 8) → HDRF (Case 7) the atmospheric correction is performed under The basic retrieval scheme starts with the assumption of a Lambertian surface. The hemispherical-conical observations (Case 8). resulting surface reﬂectances, collected during a

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Figure 15.6 Recommended processing pathway of reﬂectance products. The pathway of direct BHR retrieval from HDRF data can be performed through interpolation, thus not relying on a BRDF model (e.g., MISR, FIGOS).

period of 16 days, are subsequently used to ﬁt of direct and diffuse illumination, whereas BHRiso a BRDF model (i.e., RossThickLiSparseReciprocal (known also as white-sky albedo) involves an for MODIS (Lucht et al. 2000)). In the case of isotropic diffuse illumination only. The calculations dense angular sampling (sun and viewing geom- are based on forward modeling using the BRDF etry) and potentially measured irradiance, such model parameters as previously obtained by the as in ground level experiments, the BRDF alter- HDRF–BRDF retrieval. Only a limited selection of natively can be retrieved using radiative transfer hemispherical products are usually delivered for solutions, and thus without relying on a BRDF a particular sensor (e.g., DHR (solar angle corre- model (Martonchik 1994). Note that inverting a sponding to mean solar noon within 16 days) and BRDF model using HDRF data without previous BHRiso for MODIS, while a routine is provided to correction of the diffuse illumination may result in calculate the BHR as a linear combination of DHR a distortion of the BRDF shape in the visible and and BHRiso). For MISR, only the DHR product relies near-infrared, even for low aerosol content in the on BRDF forward modeling, representing the solar atmosphere (Lyapustin and Privette 1999). geometry at the time of observation. The MISR 3 BRDF (Case 1) → DHR (Case 3), BHR, and BHR product is directly inferred based on HDRF BHRiso (Case 9) data – both products do not involve BRDF forward The angular integration of the BRDF under differ- modeling and rely on non-isotropic illumination ent illumination conditions results in hemispheri- conditions, thus on actual diffuse and direct illu- cally integrated reﬂectance quantities. DHR (also mination corresponding to atmospheric conditions referred to as black-sky albedo) corresponds to and sun geometry at the time of observation. Thus, a direct illumination beam only, BHR (named only MISR delivers BHR products which include blue-sky albedo occasionally) to a combination direct and anisotropic diffuse illumination, most

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closely representing actual conditions. However, sensor-speciﬁc assumptions and restrictions when a modeling study showed that the assumption of integrating multiple sensor data. an isotropic diffuse illumination as compared to anisotropic diffuse illumination leads to relative albedo biases within bounds of 10% (Pinty et al. 2005). CONCLUSIONS AND RECOMMENDATIONS 4 BRDF → NBAR BRF The MODIS product suite additionally contains a This chapter presents a basic conceptual model of Nadir BRDF Adjusted Reﬂectance (NBAR) product reﬂectance terminology, complemented by exam- that is a BRF modeled for the nadir view at the ples of sensors and products in order to help the user to critically review the products of his/her mean solar zenith angle of the 16-day period. This choice, select the appropriate reﬂectance quantity means that angular effects introduced by the large and name, and process measurements according to swath width of MODIS are corrected. their physical meaning. The variety in physical quantities resulting from different sensor sampling schemes, preprocess- Several studies showed that directional and ing, atmospheric correction, and angular modeling, hemispherical illumination reﬂectance products requires a rigorous documentation standard for from current operational instruments are highly remotely sensed reﬂectance data. Beyond the algo- correlated and that the differences are generally rithm theoretical basis document with a detailed small. Schaepman-Strub et al. (2006) calculated a description of the data processing steps performed, relative difference in single-view angle products a short and standardized description on the physical (HDRF versus BRF) of up to 14%. The rela- character of the delivered reﬂectance product must tive bias of hemispherically integrated reﬂectance be accessible as well. This necessarily includes the quantities, i.e., BHR versus DHR, was smaller, and accurate listing of opening angles and directions reached a maximum of 5.1% in the blue spectral of illumination and observation, revealing whether band, under a relatively thick atmosphere (with an the product represents inherent reﬂectance proper- Aerosol Optical Depth (AOD) of 0.36 at 558 nm). ties of the surface or contains a diffuse illumination The differences generally increase with increasing component corresponding to the atmospheric and aerosol optical depth, and decrease with increas- terrain conditions of the observations. Relying on ing wavelength. On the other hand, a systematic this standardized reﬂectance description, users can numerical study of surface albedo based on the choose the appropriate reﬂectance products and radiative transfer equation for 12 land cover types evaluate whether approximations will introduce investigated the dependence on atmospheric condi- relevant biases to their applications. Numerically, tions and solar zenith angle (Lyapustin 1999). For differences between hemispherical, conical, and a large number of vegetation and soil surfaces, the directional quantities depend on various factors, range of relative variation of surface albedo with including the anisotropy of the surface, the sen- atmospheric optical depth did not exceed 10–15% ◦ sitivity distribution within the sensor IFOV, and at a solar zenith angle smaller than 50 and 20–30% ◦ ◦ its fall off outside the cone, the viewing and at solar zenith angles larger than 70 . At 52–57 sun geometry, atmospheric conditions, and the solar zenith angle the albedo is almost insensitive scattering properties of the area surrounding the to the atmospheric optical depth, resulting in a DHR observed surface. This implies that numerical dif- that is equal to the BHR. The above biases may thus ferences are wavelength dependent according to introduce a systematic error when neglected (e.g., the involved absorption and scattering processes of in vegetation indices). the atmosphere and the observed surface. The dif- The above data processing pathway illustrates fuse illumination component generally decreases that the at-surface reﬂectance quantities inferred with increasing wavelength, resulting in decreas- from satellites are not directly observed quantities, ing numerical differences from the blue toward but are based on sophisticated algorithms address- longer wavelengths. ing the atmospheric correction and the BRDF retrieval as well as forward modeling. The dif- fering sampling schemes of the observations, the applied BRDF models, as well as the atmospheric correction schemes (see, e.g., the aerosol opti- ACKNOWLEDGMENTS cal depth comparison between MODIS and MISR (Kahn et al. 2007)) introduce a bias between the This chapter is dedicated to the memory of Jim reﬂectance products of different satellite sensors, Palmer, who passed away on January 4 2007. which has not yet been assessed. It is there- The authors would like to thank Stephen Warren, fore highly recommended to select the appropriate University of Washington, for comments on usage reﬂectance product carefully, and to pay attention to of the term intensity.

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