radiometry and reﬂectance: from terminology concepts to...

[18:16 27/2/2009 5270-Warner-Ch15.tex] Paper Size: a4 paper Job No: 5270 Warner: The SAGE Handbook of Remote Sensing Page: 215 213–228

15Radiometry and Reflectance:

From Terminology Concepts toMeasured Quantities

G a b r i e l a S c h a e p m a n - S t r u b , M i c h a e lS c h a e p m a n , J o h n M a r t o n c h i k , T h o m a s

P a i n t e r , a n d S t e f a n D a n g e l

Keywords: radiometry, reflectance, terminology, reflectancemeasurements, reflectance quantities, albedo, reflectance

factor, bidirectional reflectance, BRDF.

INTRODUCTION

The remote sensing community devotes majorefforts to calibrate sensors, improve measurementsetups, and validate derived products to quan-tify and reduce measurement uncertainties. Givenrecent advances in instrument design, radiomet-ric calibration, atmospheric correction, algorithmand product development, validation, and delivery,the lack of standardization of reflectance termi-nology and products has emerged as a source ofconsiderable error.

Schaepman-Strub et al. (2006) highlighted thefact that the current use of reflectance terminol-ogy in scientific studies, applications, and pub-lications often does not comply with physicalstandards. Biases introduced by using an inappro-priate reflectance quantity can exceed minimumsensitivity levels of climate models (i.e., ±0.02reflectance units (Sellers et al. 1995)). Further,they may introduce systematic, wavelength depen-dent errors in reflectance and higher level prod-uct validation efforts, in data fusion approachesbased on different sensors, and in applications.

These differences are especially important in longterm, large area trend studies, as the latter aremostly based on multiple sensors with differentspectral and angular sampling, modeling, as wellas atmospheric correction schemes.

Optical remote sensing is based on the mea-surement of reflected and emitted electromag-netic radiation. This chapter will deal with thereflected portion of optical remote sensing. Giventhe inherent anisotropy of natural surfaces andthe atmosphere, the observed reflected radiancedepends on the actual solar zenith angle, theratio of direct to diffuse irradiance (includingits angular distribution), the observational geom-etry, including the swath width (field of view,FOV) and the opening angle of the remote sens-ing instrument (i.e., the instantaneous field of view,IFOV). Current atmospheric correction schemescompensate for the part of the observed signalwhich is contributed by the atmosphere. How-ever, these schemes mostly rely on the assump-tion of Lambertian surfaces, thus neglecting theiranisotropy and corresponding geometrical-opticaleffects introduced by the variation of illumination


216 THE SAGE HANDBOOK OF REMOTE SENSING

as well as differing IFOVs. Resulting at-surfacereflectance products may differ considerably byphysical definition further contributing to a numer-ical bias of products.

The aim of this chapter is to explain geometric-optical differences in remote sensing observationsand reflectance quantities, in order to give users thebackground to choose the appropriate product fortheir applications, to design experiments with fieldinstrumentation, and process the measurementsaccordingly.

This chapter presents a systematic and consistentdefinition of radiometric units, and a conceptualmodel for the description of reflectance quanti-ties. Reflectance terms such as BRDF, HDRF,BRF, BHR, DHR, black-sky albedo, white-skyalbedo, and blue-sky albedo are defined, explained,and exemplified, while separating conceptual frommeasurable quantities. The reflectance conceptualmodel is used to specify the measured quantitiesof current laboratory, field, airborne, and satel-lite sensors. Finally, the derivation of higher-levelreflectance products is explained, followed byexamples of operational products. All symbols andmain abbreviations used in this chapter are listedin Table 15.1.

RADIOMETRY ANDGEOMETRICAL-OPTICAL CONCEPTS

Radiometry is the measurement of optical radi-ation, which is electromagnetic radiation withinthe wavelength range 0.01–1000 micrometers(µm). Photometry follows the same definition asradiometry, except that the measured radiation isweighted by the spectral response of the humaneye. 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. Remotesensing detectors are usually not adapted to theresponse function of the human eye; therefore thischapter concentrates on radiometry. The followingtwo sections on radiometry are primarily based onan extended discussion of Palmer (2003).

Basic quantities and units - energy,power, projected area, solid angle

Radiometric units are based on two conceptualapproaches, namely those based on (a) power orenergy, or (b) geometry.

(a) Power and energy• Energy is an SI derived unit, measured in Joules,

with the recommended symbol Q.

Table 15.1 Symbols used in this chapterS distribution of direction of radiationA surface area [m2]Ap projected area [m2]E irradiance, incident flux density;

≡ d/dA[W m−2]I radiant intensity; ≡ d/dω[W sr−1]L radiance; ≡ d2

/(dA cos θdω)[W m−2sr−1]M radiant exitance, exitent flux density;

≡ d/dA[W m−2]Q energy [J]ρ reflectance; ≡ dr /di [dimensionless]R reflectance factor; ≡ dr /did

r [dimensionless]t timeβ plane angle [rad] radiant flux, power [W]θ zenith angle, in a spherical coordinate system [rad]φ azimuth angle, in a spherical coordinate

system [rad]ω solid angle; ≡ ∫

dω ≡ ∫∫sin θdθdφ[sr]

projected solid angle;≡ ∫

cos θdω ≡ ∫∫cos θ sin θdθdφ[sr]

λ wavelength of the radiation [nm]

Sub- and superscriptsi incidentr r eflectedid ideal (lossless) and d iffuse (isotropic or

Lambertian)atm atmosphericdir dir ectdiff diff use

TermsBHR BiHemispherical ReflectanceBRDF Bidirectional Reflectance Distribution FunctionBRF Bidirectional Reflectance FactorDHR Directional – Hemispherical ReflectanceHDRF Hemispherical – Directional Reflectance Factor

• Power , also known as radiant flux, is another SIderived unit. It is the derivative of energy withrespect to time, dQ/dt , and the unit is the watt (W).The recommended symbol for power is . Energyis the integral over time of power, and is used forintegrating detectors and pulsed sources, whereaspower is used for non-integrating detectors andcontinuous sources.

(b) Geometry• The projected area, Ap , is defined as the recti-

linear projection of a surface of any shape ontoa plane normal to the unit vector (Figure 15.1,top). The differential form is dAp = cos(θ) dAwhere θ is the angle between the local surfacenormal and the line of sight. The integration overthe surface area leads to Ap = ∫

Acos(θ )dA.


RADIOMETRY AND REFLECTANCE 217

Ap

S = 1

1 rad

r = 1

A = 1

1sr

r = 1

Aq

b

w

b = s/r

Figure 15.1 Top: Projected area Ap . Middle:Plane angle β. Bottom: Solid angle ω.(Reproduced with permission fromInternational Light Technologies Inc,Peabody, MA).

• The plane angle is defined as the length ofan arc (s) divided by its radius (r ), β = s/r(Figure 15.1, middle). If the arc that is subtendedby the angle is exactly as long as the radius ofthe circle, then the angle spans 1 radian. This isequivalent to about 57.2958. A full circle cov-ers an angle of 2π radians or 360, therefore theconversion between degrees and radians is 1 rad= (180/π) degrees. In SI-terminology the abovereads as follows: The radian is the plane anglebetween two radii of a circle that cuts off on thecircumference an arc equal in length to the radius(Taylor 1995).

• The solid angle, ω, extends the concept ofthe plane angle to three dimensions, and equalsthe ratio of the spherical area to the square of theradius (Figure 15.1, bottom). As an example weconsider a sphere with a radius of 1 metre. A conethat covers an area of 1 m2 on the surface of thesphere 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 57subtends a solid angle of 1 sr. In comparison, thesun covers a solid angle of only 0.00006 sr whenviewed from Earth, corresponding to a plane angleof 0.5. The projected solid angle is defined as = ∫

cos(θ )dω.

Radiometric units - irradiance, radiance,reflectance, reflectance factor, andwavelength dependence

Referring to the above concepts, we can nowapproach the basic radiometric units as used inremote sensing (Figure 15.2).

• Irradiance, E (also know as incident flux density),is measured in W m−2. Irradiance is power per unitarea incident from all directions in a hemisphereonto a surface that coincides with the base of thathemisphere (d/dA). A similar quantity is radiantexitance, M , which is power per unit area leavinga surface into a hemisphere whose base is thatsurface.

• Reflectance, ρ, is the ratio of the radiant exitance(M [W m−2]) with the irradiance (E [W m−2]), andas such dimensionless. Following the law of energyconservation, the value of the reflectance is in theinclusive interval 0 to 1.

• Radiant intensity , I , is measured in W sr−1, andis power per unit solid angle (d/dω). Note thatthe atmospheric radiation community mostly usesthe terms intensity and flux as they were definedin Chandrasekhar’s classic work (Chandrasekhar1950). More recent textbooks on atmosphericradiation still propose the use of intensity, withunits W m−2 sr−1, along with a footnote say-ing that intensity is equivalent to radiance. Anextensive discussion on the (mis-)usage of theterm intensity and corresponding units is given inPalmer (1993). He concludes that following the SIsystem definition of the base unit candela, the SIderived unit for radiant intensity is W sr−1, and forradiance W m−2 sr−1. In this chapter, we followthe SI definitions.

• Radiance, L, is expressed in the unit W m−2 sr−1,and is power per unit projected area per unit solid



m2Ap sr

srA

E

L

E = LΩL = IAp

I

Ω

ωF

Φ = Iω E = Φ/A

W

X

X

÷

÷

W

W

m2sr

m2

W

sr

m2

Figure 15.2 Overview of radiometric quantities (left) and corresponding units (right) as usedin 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 anglebetween the surface normal and the specifieddirection).

• The reflectance factor, R, is the ratio of the radi-ant flux reflected by a surface to that reflectedinto the same reflected-beam geometry and wave-length range by an ideal (lossless) and diffuse(Lambertian) standard surface, irradiated underthe same conditions. Reflectance factors can reachvalues beyond 1, especially for strongly forwardreflecting surfaces such as snow (Painter andDozier 2004). For measurement purposes, a Spec-tralon panel commonly approximates the idealdiffuse standard surface. This is a manufacturedstandard having a high and stable reflectancethroughout the optical region, approximates aLambertian surface, and is traceable to the U.S.National Institute of Standards and Technology(NIST). Its use in field spectroscopy and additionalreferences are described in Milton et al. (in press).We assume further that an isotropic behaviorimplies a spherical source that radiates the samein all directions, i.e., the intensity [W sr−1] is thesame in all directions. The Lambertian behaviorrefers to a flat reflective surface. The intensityof light reflected from a Lambertian surface fallsoff as the cosine of the observation angle withrespect to the surface normal (Lambert’s cosinelaw), whereas the radiance L [W m−2 sr−1] isindependent of observation angle. The reflectedradiant flux from a given area is reduced bythe cosine of the observation angle, but theobserved area has increased by the cosine of theangle, and therefore the observed radiance is thesame independent of observation angle. Note that

Lambertian always refers to a flat surface withthe reflected intensity falling off as the cosineof the observation angle with respect to the sur-face normal (Lambert’s cosine law). Isotropic onthe other hand means ‘having the same prop-erties in all directions’, and does not refer to aspecific physical quantity. Therefore, a perfectlydiffuse or Lambertian surface element dA is onefor which the reflected radiance is isotropic, withthe same value for all directions into the full hemi-sphere above the element dA of the reflectingsurface.

Geometrical-optical concepts –directional, conical, and hemispherical

The anisotropic reflectance properties of a sur-face (Figure 15.3) can mathematically be describedby the bidirectional reflectance distribution func-tion (BRDF). The term bidirectional implies singledirections for the incident and reflected radiances(entering and emanating from differential solidangles, respectively). This mathematical conceptcan only be approximated by measurements, sinceinfinitesimal elements of solid angle do not includemeasurable amounts of radiant flux (Nicodemuset al. 1977), and unlimited small light sources, aswell as an unlimited small sensor instantaneousfield of view (IFOV) do not exist. Consequently,all measurable quantities of reflectance are per-formed either in the conical or hemisphericaldomain. From a physical point of view, we there-fore differentiate between conceptual (directional)and measurable quantities (involving conical andhemispherical solid angles of observation and illu-mination). According to Nicodemus et al. (1977),



Figure 15.3 Reflectance anisotropy of avegetation canopy showing the dependenceof the observed reflectance on the viewingdirection and IFOV of the remote sensinginstrument (black). The shape of thereflectance distribution changes with solarangle and ratio of direct solar radiation anddiffuse (radiation scattered by theatmosphere) illumination. It can also be seenthat vegetation is mainly a backwardscattering object, with a so-calledreflectance hot spot toward the mainillumination direction (as opposed to snow,which is a forward scatterer), while waxyleaves may introduce a forward scatteringcomponent.

the angular characteristics of the incoming radianceare named first in the reflectance term, followed bythe angular characteristics of the reflected radiance.This results in nine different cases of reflectancequantities, illustrated in Figure 15.4. The math-ematical derivations for the different cases aregiven below, followed by sections elaborating thegeometrical configurations of most common mea-surement setups and on the resulting recommendedterminology.

REFLECTANCE QUANTITIES IN REMOTESENSING – BRDF, BRF, HDRF, DHR, BHR

Based on the above concepts, we can developthe corresponding mathematical formula-tions for the most relevant quantities usedin remote sensing, namely the BRDF (Bidi-rectional Reflectance Distribution Function),BRF (Bidirectional Reflectance Factor), HDRF(Hemispherical-Directional Reflectance Factor),DHR (Directional-Hemispherical Reflectance),and BHR (Bihemispherical Reflectance). The sameconcepts may be extended to the transmittancebehavior (BTDF, Bidirectional Transmittance

Distribution Function), for example whenmeasuring and modeling leaf optical properties.

We symbolize reflectance and reflectancefactor as

ρ(Si, Sr, λ) = reflectance, and

R (Si, Sr, λ) = reflectance factor

where Si and Sr describe the angular distributionof all incoming and reflected radiation observedby the sensor, respectively. Si and Sr only describea set of angles occurring with the incoming andreflected radiation and not their intensity distribu-tions. Sr represents a cone with a given solid anglecorresponding to a sensor’s instantaneous field ofview (IFOV), but no sensor weight functions areincluded here. This becomes only necessary if thesensitivity of the sensor depends on the locationwithin the rim of the cone. When a sensor has a dif-ferent IFOV for different wavelength ranges, thenSr depends on the wavelength.

The terms Si and Sr can be expanded into a moreexplicit angular notation to address the remotesensing problem:

ρ (θi, φi, ωi; θr, φr, ωr; λ), and

R (θi, φi, ωi; θr, φr, ωr; λ)

where the directions (θ and φ are the zenith andazimuth angle, respectively) of the incoming (sub-script i) and the reflected (subscript r) radiation,and the associated solid angles of the cones (ω) areindicated. This notation follows the definition of ageneral cone.

For surface radiation measurements made fromspace, aircraft or on the ground, under ambient skyconditions, the cone of the incident radiation isof hemispherical extent (ω = 2π[sr]). The inci-dent radiation may be divided into a direct sunlightcomponent and a second component, namely sun-light which has been scattered by the atmosphere,the terrain, and surrounding objects, resulting in ananisotropic, diffuse illumination, sometimes called‘skylight’.

The above reflectance and reflectance factordefinitions lead to the following special cases:

• ωi or ωr are omitted when either is zero (direc-tional quantities).

• If 0 < (ωi or ωr ) < 2π , then θ, φ describe thedirection of the center axis of the cone (e.g., theline from a sensor to the center of its ground fieldof view – conical quantities).

• If ωi = 2π , the angles θi , φi indicate the direc-tion of the incoming direct radiation (e.g., the posi-tion of the sun). For remote sensing applications,it is often useful to separate the natural incomingradiation into a direct (neglecting the sun’s size)



Incoming/Reflected Directional Conical Hemispherical

Directional BidirectionalCase 1

Directional-conicalCase 2

Directional-hemisphericalCase 3

Conical Conical-directionalCase 4

BiconicalCase 5

Conical-hemisphericalCase 6

Hemispherical Hemispherical-directionalCase 7

Hemispherical-conicalCase 8

BihemisphericalCase 9

Figure 15.4 Geometrical-optical concepts for the terminology of at-surface reflectancequantities. All quantities including a directional component (i.e., Cases 1–4, 7) are conceptualquantities, whereas measurable quantities (Cases 5, 6, 8, 9) are shaded in gray. (Reprintedfrom 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. RemoteSensing of Environment, 103: 27–42). (See the color plate section of this volume for a colorversion of this figure.)

and hemispherical diffuse part. One may alsoinclude a terrain reflected diffuse component thatis calculated with a topographic radiation modelsuch as TOPORAD (Dozier 1980). Consequently,the preferred notation for the geometry of theincoming radiation under ambient illuminationconditions is θi , φi , 2π . Note that in this case,θi , φi describe the position of the sun and not thecenter of the cone (2π ). In the case of an isotropicdiffuse irradiance field, without any direct irradi-ance component (closest approximated in the caseof an optically thick cloud deck), θi , φi are omit-ted. Isotropic behavior implies that the intensity[W sr−1] is the same in all directions.

• If ωr = 2π , θr and φr are omitted.

It should be noted that the nine standardreflectance terms defined by (Nicodemus et al.1977) ‘are applicable only to situations with uni-form and isotropic radiation throughout the inci-dent beam of radiation’. They then state that,‘If this is not true, then one must refer to themore general expressions’. This implies that anysignificant change to the nine reflectance conceptswhen the incident radiance is anisotropic lies in themathematical expression used in their definition.

Based on this implication, Martonchik et al., 2000,adapted the terminology to the remote sensing case,which involves direct and diffuse sky illumina-tion. In the following, we give the mathematicaldescription of the most commonly used quantitiesin remote sensing, thus the general expressions fornon-isotropic incident radiation. When applicable,we simplify the expression for the special case ofisotropic incident radiation. Further, the particularwavelength dependency is omitted as well in mostcases to improve readability of the equations. How-ever, it must be understood that all interaction oflight with matter is wavelength dependent, and maynot simply be ignored.

The bidirectional reflectancedistribution function (BRDF) – Case 1

The bidirectional reflectance distribution function(BRDF) describes the scattering of a parallel beamof incident light from one direction in the hemi-sphere into another direction in the hemisphere.The term BRDF was first used in the literature in theearly 1960s (Nicodemus 1965). Being expressedas the ratio of infinitesimal quantities, it cannot bedirectly measured (Nicodemus et al. 1977). The



BRDF describes the intrinsic reflectance proper-ties of a surface and thus facilitates the derivationof many other relevant quantities, e.g., conicaland hemispherical quantities, by integration overcorresponding finite solid angles.

The spectral BRDF, fr(θi, φi; θr, φr; λ) can beexpressed as:

BRDFλ = fr (θi, φi; θr , φr; λ)

= dLr (θi, φi; θr , φr; λ)

dEi(θi, φi; λ)[sr−1]. (1)

For reasons of clarity, we will omit the spectraldependence in the following. We therefore writefor the BRDF:

BRDF = fr (θi, φi; θr , φr )

= dLr (θi, φi; θr , φr )

dEi(θi, φi)[sr−1]. (2)

Reflectance factors – Definition ofCases 1, 5, 7 and 8

When reflectance properties of a surface are mea-sured, the procedure usually follows the definitionof a reflectance factor. The reflectance factor is theratio of the radiant flux reflected by a sample sur-face to the radiant flux reflected into the identicalbeam geometry by an ideal (lossless) and diffuse(Lambertian) standard surface, irradiated under thesame conditions as the sample surface.

The bidirectional reflectance factor (BRF;Case 1) is given by the ratio of the reflected radiantflux from the surface area dA to the reflected radiantflux from an ideal and diffuse surface of the samearea dA under identical view geometry and singledirection illumination:

BRF = R(θi, φi; θr , φr ) = dr (θi, φi; θr , φr )

didr (θi, φi)

(3)

= cos θr sin θrdLr (θi, φi; θr , φr )dθrdφrdA

cos θr sin θrdLidr (θi, φi)dθrdφrdA

(4)

= dEi(θi, φi)

dLidr (θi, φi)

· dLr (θi, φi; θr , φr )

dEi(θi, φi)(5)

= fr (θi, φi, θr , φr )

f idr (θi, φi)

= π fr (θi, φi; θr , φr ). (6)

An ideal Lambertian surface reflects the sameradiance in all view directions, and its BRDFis 1/π . Thus, the BRF [unitless] of any surfacecan be expressed as its BRDF [sr−1] times π(Equation (6)). For id

r and Lidr , we omit the

view zenith and azimuth angles, because there isno angular dependence for the ideal Lambertiansurface.

The concept of the hemispherical-directionalreflectance factor (HDRF; Case 7) is similar to

the definition of the BRF, but includes irradiancefrom the entire hemisphere. This makes the quan-tity dependent on the actual, simulated or assumedatmospheric conditions and the reflectance of thesurrounding terrain. This includes spectral effectsintroduced by the variation of the diffuse to directirradiance ratio with wavelength (e.g., Strub et al.2003).

HDRF

= R(θi, φi, 2π; θr , φr ) = dr (θi, φi, 2π ; θr , φr )

didr (θi, φi, 2π )

(7)

= cos θr sin θrLr (θi, φi, 2π ; θr , φr )dθrdφrdA

cos θr sin θrLidr (θi, φi, 2π )dθrdφrdA

(8)

= Lr (θi, φi, 2π ; θr , φr )

Lidr (θi, φi, 2π )

=∫

2πfr (θi, φi; θr , φr )di(θi, φi)∫

2π(1/π )di(θi, φi)

(9)

=∫ 2π

0

∫ π/20 fr (θi, φi; θr , φr ) cos θi sin θiLi(θi, φi)dθidφi

(1/π )∫ 2π

0

∫ π/20 cos θi sin θiLi(θi, φi)dθidφi

.

(10)

If we divide Li into a direct (Edir with anglesθ0, φ0) and diffuse part, we may continue:

=

⎛⎜⎝

fr (θ0,φ0;θr ,φr )Edir (θ0,φ0)

+2π∫0

π/2∫0

fr (θi,φi;θr ,φr )cosθi sinθiLdiffi (θi,φi)dθidφi

⎞⎟⎠

⎛⎜⎝

(1/π )(Edir (θ0,φ0)

+2π∫0

π/2∫0

cosθi sinθiLdiffi (θi,φi)dθidφi)

⎞⎟⎠

(11)

then, if and only if Ldiffi is isotropic (i.e., indepen-

dent of the angles), we may continue:

=

⎛⎜⎝

fr (θ0, φ0; θr , φr )Edir (θ0, φ0)

+Ldiffi

2π∫0

π/2∫0

fr (θi, φi; θr , φr ) cos θi sin θidθidφi

⎞⎟⎠

((1/π )

[Edir (θ0, φ0) + Ldiff

i

2π∫0

π/2∫0

cos θi sin θidθidφi

] )

(12)

= π fr (θ0, φ0; θr , φr )(1/π )Edir (θ0, φ0)

(1/π )Edir (θ0, φ0) + Ldiffi

+∫ 2π

0

∫ π/2

0fr (θi, φi; θr , φr ) cos θi sin θidθidφi

× Ldiffi

(1/π )Edir (θ0, φ0) + Ldiffi

(13)

= R(θ0, φ0; θr , φr )d + R(2π ; θr , φr )(1 − d) (14)

where d corresponds to the fractional amount ofdirect radiant flux (i.e., d∈[0, 1]).



The biconical reflectance factor (conical–conical reflectance factor, CCRF; Case 5), isdefined as:

CCRF

= R(θi, φi, ωi; θr , φr , ωr )

=

∫ωr

∫ωi

fr (θi, φi; θr , φr )Li(θi, φi)didr(r/π) ∫ωi

Li(θi, φi)di(15)

where = ∫d = ∫

cosθdω = ∫ ∫cosθ sin θ

dθdφ is the projected solid angle of the cone.Formally, the CCRF can be seen as the most gen-

eral quantity, because its expression contains allother cases as special ones: for ω = 0 the inte-gral collapses and we obtain the directional case,and for ω = 2π we obtain the hemispherical case.However, the BRF and BRDF remain the most fun-damental and desired quantities because they arethe only quantities not integrated over a range ofangles.

For large IFOV measurements performed underambient sky illumination, the assumption of a zerointerval of the solid angle for the measured reflectedradiance beam does not hold true. The result-ing quantity most precisely could be described ashemispherical-conical reflectance factor (HCRF;Case 8), obtained from Equation (15) by settingωi = 2π :

HCRF = R(θi, φi, 2π ; θr , φr , ωr )

=

∫ωr

∫2π

fr (θi, φi; θr , φr )Li(θi, φi)didr(r/π) ∫

2π

Li(θi, φi)di. (16)

Reflectance – Definition of Cases 3 and 9

When applying remote sensing observations tosurface energy budget studies, for example, thetotal energy reflected from a surface is of inter-est, rather than a reflectance quantity directedinto a small solid angle. In the following, wedescribe the hemispherical reflectance as a func-tion of different irradiance scenarios including(i) the special condition of pure direct irradiance,(ii) common Earth irradiance, composed of dif-fuse and direct components, and (iii) pure diffuseirradiance.

The directional-hemispherical reflectance(DHR; Case 3) corresponds to pure direct illumi-nation (reported as black-sky albedo in the MODIS(Moderate Resolution Imaging Spectroradiometer)product suite (Lucht et al. 2000)). It is the ratio ofthe radiant flux for light reflected by a unit surfacearea into the view hemisphere to the illuminationradiant flux, when the surface is illuminated

with a parallel beam of light from a singledirection.

DHR

= ρ(θi, φi; 2π ) = dr (θi, φi; 2π )

di(θi, φi)

= dA∫ 2π

0∫ π/2

0 dLr (θi, φi; θr , φr ) cos θr sin θr dθr dφr

di(θi, φi)(17)

= di(θi, φi)∫ 2π

0∫ π/2

0 fr (θi, φi; θr , φr ) cos θr sin θr dθr dφr

di(θi, φi)(18)

=∫ 2π

0

∫ π/2

0fr (θi, φi; θr , φr ) cos θr sin θr dθr dφr . (19)

The bihemispherical reflectance (BHR; Case 9),generally called albedo, is the ratio of the radi-ant flux reflected from a unit surface area into thewhole hemisphere to the incident radiant flux ofhemispherical angular extent:

BHR

= ρ(θi, φi, 2π ; 2π ) = dr (θi, φi, 2π ; 2π )

di(θi, φi, 2π )(20)

= dA∫ 2π

0∫ π/2

0 dLr (θi, φi, 2π ; θr , φr ) cos θr sin θr dθr dφr

dA∫ 2π

0∫ π/2

0 dLi(θi, φi) cos θi sin θidθidφi

(21)

=

⎛⎜⎝

2π∫0

π/2∫0

2π∫0

π/2∫0

fr (θi, φi; θr , φr ) cos θr sin θr dθr dφr

Li(θi, φi) cos θi sin θidθidφi

⎞⎟⎠

2π∫0

π/2∫0

Li(θi, φi) cos θi sin θidθidφi

(22)

=∫ 2π

0∫ π/2

0 ρ(θi, φi; 2π )Li(θi, φi) cos θi sin θidθidφi∫ 2π0

∫ π/20 Li(θi, φi) cos θi sin θidθidφi

. (23)

If as before we divide Li into a direct (Edir withangles θ0, φ0) and diffuse part, and assume thatLdiff

i is isotropic we can write:

=

⎛⎜⎜⎝

ρ(θ0, φ0; 2π )Edir (θ0, φ0)

+π Ldiffi (1/π )

2π∫0

π/2∫0

ρ(θi, φi; 2π ) cos θi sin θidθidφi

⎞⎟⎟⎠

Edir (θ0, φ0) + π Ldiffi

(24)

= ρ(θ0, φ0; 2π )d + ρ(2π; 2π )(1 − d) (25)

where d again corresponds to the fractional amountof direct radiant flux.

For the special case of pure diffuse isotropic inci-dent radiation, a situation that may be most closelyapproximated in the field by a thick cloud or aerosollayer, the resulting BHR (reported as white-skyalbedo in the MODIS product suite (Lucht et al.2000), and sometimes also referred to as BHRiso)



can be described as follows:

BHR=ρ(2π;2π )

= 1

π

∫ 2π

0

∫ π/2

0ρ(θi,φi;2π )cosθi sinθidθidφi.

(26)

Under ambient illumination conditions, thealbedo is influenced by the combined diffuse anddirect irradiance. To obtain an approximation of thealbedo for ambient illumination conditions (alsoreported as blue-sky albedo in the MODIS productsuite), it is suggested that the BHR for isotropicdiffuse illumination conditions and the DHR becombined linearly (see Equation (25)), correspond-ing to the actual ratio of diffuse to direct illumina-tion (Lewis and Barnsley 1994, Lucht et al. 2000).The diffuse component then can be expressed as afunction of wavelength, optical depth, aerosol type,and terrain contribution. The underlying assump-tion of an isotropic diffuse illumination may lead tosignificant uncertainties due to ignoring the actualdistribution of the incoming diffuse radiation (e.g.,Pinty et al. 2005).

All the above-mentioned albedo values, with theexception of the BHR for pure diffuse illuminationconditions, depend on the actual illumination angleof the direct component. Thus it is highly recom-mended to include the illumination geometry in themetadata of albedo quantities.

OBSERVATIONAL GEOMETRY OF REMOTESENSING INSTRUMENTS

This section discusses the geometric configurationof selected operational sensors, including labo-ratory and field instruments, as well as airborneand spaceborne sensors, using the basic concep-tual model as presented in Figure 15.4. Froma strict physical point of view, the most com-mon measurement setup of satellites, airborne, andfield instruments corresponds to the hemispherical-conical configuration (Case 8), while laboratoryconditions are mostly biconical (Case 5).

Laboratory instruments

Laboratory conditions provide the ability tomeasure reflectance properties under controlledenvironmental conditions, being independent ofirradiance variations due to a changing atmosphere,time of the day, or season. This is desirable wheninherent reflectance properties (i.e., the BRDF) of asurface are investigated. Laboratory measurementsinvolve an artificial light source, which is usually

non-parallel (due to internal beam divergence andcollimating limitations), whereas solar direct illu-mination can be approximated as being parallel(within 0.5). The diffuse illumination compo-nent in the laboratory can be minimal when thereflections are minimized (e.g., walls are paintedblack and black textiles cover reflecting objects).For the Laboratory Goniometer System (LAGOS),the diffuse-to-total illumination ratio was shownto be lower than 0.5% in the spectral range of400–1000 nm (Dangel et al. 2005). The instanta-neous field of view (IFOV) of a few degrees of thenon-imaging spectroradiometer employed corre-sponds to a conical opening angle. Given the aboveconditions, the typical measurement setup of labo-ratory spectrometer measurements corresponds tothe biconical configuration, resulting in conical–conical reflectance factors (CCRF – Case 5). For aperfectly collimated light source and a small IFOV,measurements may approximate the bidirectionalquantity (e.g., the SpectroPhotoGoniometer (SPG)to measure leaf optical properties (Combes et al.2007)).

Ground based field instruments

In the field, ambient illumination always includes adiffuse fraction. Its magnitude and angular distribu-tion depend on the actual atmospheric conditions,surrounding terrain and objects, and wavelength.Thus, outdoor measurements always include hemi-spherical illumination, which can be describedas a composition of a direct and an anisotropicdiffuse component. Shading experiments are dis-cussed in Schaepman-Strub et al. (2006), where itis concluded that they are only suitable to sepa-rate direct and diffuse illumination if the shadingobject exactly covers the solar disc (0.00006 sr).The reason is that a significant fraction of diffuseillumination is located within a small cone in thedirection of the direct illumination of the sun.

The partitioning into direct and diffuse illu-mination influences the radiation regime withinvegetation canopies. Based on a modeling study,Alton (2007) showed that the light use effi-ciency (LUE) of three forest canopies increases by6–33% when the irradiance is dominated by dif-fuse rather than direct sunlight. This demonstratesthe importance of accompanying field spectrometercampaigns with sun photometer measurements toassess the contribution of direct and diffuse irra-diance. For field instruments with an IFOV fullcone angle of about 4–5 degrees (e.g., PARABOLA(Portable Apparatus for Rapid Acquisition of Bidi-rectional Observation of the Land andAtmosphere,Abdou et al. 2001) or ASG (Automated Spectro-Goniometer, Painter et al. 2003)), the surface direc-tional reflectance variability across the opening



angle needs to be investigated. As long as thisvariability is unknown or not neglectable and cor-rected for, the measurements should be reportedas HCRF (Case 8). This is especially true forsensors with larger IFOV, such as the ASD (Ana-lytical Spectral Devices) FieldSpec series (25,while fore-optics allow a restriction to 8 or less),and in cases where the sensitivity of the sen-sor outside of the cone only gradually falls offacross several degrees outside of the half powerpoint. More details concerning field spectrome-ter measurements can be found in Milton et al.(in press).

Albedometers are designed to cover the fulldown- and upward hemisphere (two pyranometerswith an IFOV of 180 each) and approximate thebihemispherical configuration (Case 9) (e.g., Kippand Zonen 2000).

Airborne sensors

The surface illumination conditions for airbornesensor observations are the same as for fieldmeasurements, thus of hemispherical extent (seeabove). The IFOV of airborne sensors is usuallyvery 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-ital Airborne Imaging Spectrometer 7915 (DAIS7915), and 0.129 for the HyMap airbornehyperspectral scanner. In a strict physical sense,airborne observations therefore correspond tothe hemispherical-conical configuration (HCRF –Case 8), while numerically approaching thehemispherical-directional configuration (HDRF –Case 7). Most correction schemes for airborne datado not correct for the hemispherical irradiance andthus the resulting at-surface reflectances approx-imately correspond to HDRFs (Schaepman-Strubet al. 2006).

Satellite sensors

Ambient illumination conditions of hemispheri-cal extent are also present at the Earth surfacewhen observed from spaceborne sensors. Gen-erally, space-based instruments with a spatialresolution of about 1 km have an IFOV witha full cone angle of approximately 0.1 (e.g.,Multiangle Imaging SpectroRadiometer (MISR),MODerate resolution Imaging Spectroradiome-ter (MODIS), Advanced Very High ResolutionRadiometer (AVHRR)). If the HDRF is constantover the full cone angle of the instrument IFOV,then the HCRF numerically equals the HDRF. Thisapproximation is mostly used when processingsatellite sensor data.

Multi-angular sampling principles

All approaches correcting for the bias introducedby varying sun and view angles rely on multi-angular information to infer the BRDF as intrinsicreflectance property of the surface. The BRDF is afunction of the solar and observational angles, thusmeasurements are performed under changing illu-mination or viewing geometries or a combinationof both. Instantaneous multi-angular sampling isvery rare, as most sampling schemes rely on tiltingsensors and thus changing their viewing geometry.The MISR satellite has nine cameras with fixedviewing angles, and approximately 7 min lapsesbetween the first and the last camera overpass for aselected area. For non-instantaneous multi-angularmeasuring concepts, assumptions on the temporalstability of the surface or the atmospheric compo-sition (e.g., aerosol optical depth) are often made.This is a disadvantage for highly variable surfacessuch as vegetation canopies which change theirphysiological state throughout a day, or snowmeltevents lasting several days. A selection of the mostcommon multi-angular sampling principles of lab-oratory, field, airborne, and satellite sensors is givenin Figure 15.5.

PROCESSING OF REFLECTANCEPRODUCTS

While the preceding section explained the observa-tion geometry of operational sensors and the multi-angular sampling principles, this section will focuson the derivation of various reflectance quantitiesfrom the observations.

Most state of the art atmospheric correctionschemes convert top of atmosphere radiance to onesingular view angle at-surface reflectance, whilepreserving the influence of the diffuse illumina-tion on the surface reflectance, thus representingHDRF data. However, this reflectance quantitydoes not exactly represent what is required formany applications, such as energy budget studies,multi-temporal investigations, and studies rely-ing on multiple sensor data. The main productpathways to obtain higher level reflectance prod-ucts are discussed below, namely (a) removingthe effect of the diffuse hemispherical illumina-tion in single view angle observations to obtaininherent reflectance properties of the surface (i.e.,the BRDF), (b) interpolating and extrapolating thesingle-angle observations to the entire reflectedhemisphere to obtain albedo quantities, and (c)normalizing the single-angle observations to astandardized viewing geometry (i.e., to computeNadir BRDF Adjusted Reflectance, NBAR). Allthree approaches are based on the derivation ofthe BRDF, thus on the extraction of the intrinsic



Figure 15.5 Examples of multi-angular sampling principles: (a) laboratory facility to measureleaf optical properties (SPG) (photo courtesy of Stéphane Jacquemoud), (b) field goniometersystem (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-angularsampling (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., MODIS16 days). (See the color plate section of this volume for a color version of this figure.)

reflectance properties of the surface, using multi-angular sampling of the observations through vari-ation of sun and/or viewing angles. The derivationof the BRDF based on laboratory measurementsrequires a correction for conicity and inhomo-geneity of the artificial illumination source (fordetails see Dangel et al. 2005). The derivationof different at-surface reflectance quantities frommeasurements usually requires a sophisticated pro-cessing scheme (Figure 15.6), as for exampleimplemented for the MISR surface reflectanceproducts (Martonchik et al. 1998). Unfortunately,this issue does not always receive sufficient atten-tion in remote sensing when implementing process-ing schemes for ‘reflectance’ products. Below, themain required processing steps to infer the wholesuite of reflectance quantities are described, includ-ing some examples as implemented in operationalalgorithms, and the assumptions used. For eachheading, the input reflectance quantity of the pro-cessing scheme is specified on the left hand side ofthe arrow, and the resulting reflectance quantity onthe right.

1 HCRF (Case 8) → HDRF (Case 7)The basic retrieval scheme starts withhemispherical-conical observations (Case 8).

Currently, most of the existing processingapproaches assume that the HDRF is constantover the full cone angle of the instrumentIFOV, thus the HCRF numerically equals theHDRF (Case 7) without further correction. Givena sufficient number of viewing angles (e.g.,MISR), the BHR is directly derived throughinterpolation. The algorithm for retrieving theHDRF and BHR from MISR top-of-atmosphere(TOA) radiances is virtually independent of anyparticular kind of surface BRF model and itsaccuracy mainly depends on the accuracy of theatmospheric information used (Martonchik et al.1998).

2 HDRF (Case 7) → BRDF and BRF (Case 1)BRF data are derived using a parameterizedBRDF model to eliminate the diffuse illumina-tion effects present in the HDRFs (e.g., Modi-fied Rahman Pinty Verstraete (MRPV) for MISR(Martonchik et al. 1998) and ground based mea-surements (Lyapustin and Privette 1999)). Analternative approach is used for MODIS, wherethe atmospheric correction is performed underthe assumption of a Lambertian surface. Theresulting surface reflectances, collected during a



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

period of 16 days, are subsequently used to fita BRDF model (i.e., RossThickLiSparseReciprocalfor MODIS (Lucht et al. 2000)). In the case ofdense angular sampling (sun and viewing geom-etry) and potentially measured irradiance, suchas in ground level experiments, the BRDF alter-natively can be retrieved using radiative transfersolutions, and thus without relying on a BRDFmodel (Martonchik 1994). Note that inverting aBRDF model using HDRF data without previouscorrection of the diffuse illumination may result ina distortion of the BRDF shape in the visible andnear-infrared, even for low aerosol content in theatmosphere (Lyapustin and Privette 1999).

3 BRDF (Case 1) → DHR (Case 3), BHR, andBHR iso (Case 9)The angular integration of the BRDF under differ-ent illumination conditions results in hemispheri-cally integrated reflectance quantities. DHR (alsoreferred to as black-sky albedo) corresponds toa direct illumination beam only, BHR (namedblue-sky albedo occasionally) to a combination

of direct and diffuse illumination, whereas BHRiso(known also as white-sky albedo) involves anisotropic diffuse illumination only. The calculationsare based on forward modeling using the BRDFmodel parameters as previously obtained by theHDRF–BRDF retrieval. Only a limited selection ofhemispherical products are usually delivered fora particular sensor (e.g., DHR (solar angle corre-sponding to mean solar noon within 16 days) andBHRiso for MODIS, while a routine is provided tocalculate the BHR as a linear combination of DHRand BHRiso). For MISR, only the DHR product relieson BRDF forward modeling, representing the solargeometry at the time of observation. The MISRBHR product is directly inferred based on HDRFdata – both products do not involve BRDF forwardmodeling and rely on non-isotropic illuminationconditions, thus on actual diffuse and direct illu-mination corresponding to atmospheric conditionsand sun geometry at the time of observation. Thus,only MISR delivers BHR products which includedirect and anisotropic diffuse illumination, most



closely representing actual conditions. However,a modeling study showed that the assumption ofan isotropic diffuse illumination as compared toanisotropic diffuse illumination leads to relativealbedo biases within bounds of 10% (Pinty et al.2005).

4 BRDF → NBAR BRFThe MODIS product suite additionally contains aNadir BRDF Adjusted Reflectance (NBAR) productthat is a BRF modeled for the nadir view at themean solar zenith angle of the 16-day period. Thismeans that angular effects introduced by the largeswath width of MODIS are corrected.

Several studies showed that directional andhemispherical illumination reflectance productsfrom current operational instruments are highlycorrelated and that the differences are generallysmall. Schaepman-Strub et al. (2006) calculated arelative difference in single-view angle products(HDRF versus BRF) of up to 14%. The rela-tive bias of hemispherically integrated reflectancequantities, i.e., BHR versus DHR, was smaller, andreached a maximum of 5.1% in the blue spectralband, under a relatively thick atmosphere (with anAerosol Optical Depth (AOD) of 0.36 at 558 nm).The differences generally increase with increasingaerosol optical depth, and decrease with increas-ing wavelength. On the other hand, a systematicnumerical study of surface albedo based on theradiative transfer equation for 12 land cover typesinvestigated the dependence on atmospheric condi-tions and solar zenith angle (Lyapustin 1999). Fora large number of vegetation and soil surfaces, therange of relative variation of surface albedo withatmospheric optical depth did not exceed 10–15%at a solar zenith angle smaller than 50 and 20–30%at solar zenith angles larger than 70. At 52–57solar zenith angle the albedo is almost insensitiveto the atmospheric optical depth, resulting in a DHRthat is equal to the BHR. The above biases may thusintroduce a systematic error when neglected (e.g.,in vegetation indices).

The above data processing pathway illustratesthat the at-surface reflectance quantities inferredfrom satellites are not directly observed quantities,but are based on sophisticated algorithms address-ing the atmospheric correction and the BRDFretrieval as well as forward modeling. The dif-fering sampling schemes of the observations, theapplied BRDF models, as well as the atmosphericcorrection schemes (see, e.g., the aerosol opti-cal depth comparison between MODIS and MISR(Kahn et al. 2007)) introduce a bias between thereflectance products of different satellite sensors,which has not yet been assessed. It is there-fore highly recommended to select the appropriatereflectance product carefully, and to pay attention to

sensor-specific assumptions and restrictions whenintegrating multiple sensor data.

CONCLUSIONS AND RECOMMENDATIONS

This chapter presents a basic conceptual model ofreflectance terminology, complemented by exam-ples of sensors and products in order to help theuser to critically review the products of his/herchoice, select the appropriate reflectance quantityand name, and process measurements according totheir physical meaning.

The variety in physical quantities resulting fromdifferent sensor sampling schemes, preprocess-ing, atmospheric correction, and angular modeling,requires a rigorous documentation standard forremotely sensed reflectance data. Beyond the algo-rithm theoretical basis document with a detaileddescription of the data processing steps performed,a short and standardized description on the physicalcharacter of the delivered reflectance product mustbe accessible as well. This necessarily includes theaccurate listing of opening angles and directionsof illumination and observation, revealing whetherthe product represents inherent reflectance proper-ties of the surface or contains a diffuse illuminationcomponent corresponding to the atmospheric andterrain conditions of the observations. Relying onthis standardized reflectance description, users canchoose the appropriate reflectance products andevaluate whether approximations will introducerelevant biases to their applications. Numerically,differences between hemispherical, conical, anddirectional quantities depend on various factors,including the anisotropy of the surface, the sen-sitivity distribution within the sensor IFOV, andits fall off outside the cone, the viewing andsun geometry, atmospheric conditions, and thescattering properties of the area surrounding theobserved surface. This implies that numerical dif-ferences are wavelength dependent according tothe involved absorption and scattering processes ofthe atmosphere and the observed surface. The dif-fuse illumination component generally decreaseswith increasing wavelength, resulting in decreas-ing numerical differences from the blue towardlonger wavelengths.

ACKNOWLEDGMENTS

This chapter is dedicated to the memory of JimPalmer, who passed away on January 4 2007.

The authors would like to thank Stephen Warren,University of Washington, for comments on usageof the term intensity.



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