15 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 [18:16 27/2/2009 5270-Warner-Ch15.tex] Paper Size: a4 paper Job No: 5270 Warner: The SAGE Handbook of Remote Sensing Page: 215 213–228 216 THE SAGE HANDBOOK OF REMOTE SENSING 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 [18:16 27/2/2009 5270-Warner-Ch15.tex] Paper Size: a4 paper Job No: 5270 Warner: The SAGE Handbook of Remote Sensing Page: 216 213–228 RADIOMETRY AND REFLECTANCE 217 • 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.