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1 Properties of Radiation Lecture outline Flux and intensity Solid angle and the steradian Inverse square law Global insolation Interaction of radiation with matter

2 Flux or Flux density Flux (or flux density), F: rate at which radiation is incident on, or passes through, a flat surface (e.g., the ground, the top of a cloud layer, a level in the atmosphere ); Units: W m -2 By definition, a broadband quantity integrated between wavelength limits (λ 1 and λ 2 ) Spectral flux (or monochromatic flux ): flux contributed by radiation over some narrow wavelength interval; Units: W m -2 µm -1 e.g., the incident flux of solar radiation on an area of Earth s surface No information on direction of origin

3 Intensity or radiance Radiant intensity or radiance, I: describes both the strength and direction of radiation sources contributing to the flux incident on a surface; Units: W m -2 sr -1 (Watts per square meter per steradian) Remote sensing instruments, e.g., a satellite sensor viewing a narrow range of directions, measure radiance Spectral radiance: intensity measured over some narrow wavelength interval; Units: W m -2 sr -1 µm -1 Roughly corresponds to brightness of a radiation source e.g., the sky, clouds, the Sun.

4 Spherical polar coordinates θ = zenith angle, usually relative to local vertical θ = 0º is overhead, θ = 90º is the horizon. Φ = azimuth angle, e.g., the direction of the sun Direction plays an important role in any discussion of radiation

5 Solid angle: the steradian How much of the visual field of view is occupied by an object? 1 steradian Three-dimensional analog of the planar radian (ratio of arc length:radius) Ω = area of surface A / r 2 (NB. Solid angle is dimensionless) Sphere subtends a solid angle of sr The entire sky (hemisphere) above the horizon subtends sr Sun seen from Earth covers ~ sr

6 Solid angle: the steradian sin θ accounts for the convergence of longitude lines at the pole dω = sinθ dθ dφ = 2π sinθ dθ = 4π 4π 2π 0 π 0 π 0

7 Solid angle problem Mean distance of the moon from Earth = 3.84 x 10 5 km Radius of the moon = 1.74 x 10 3 km Mean distance of the Sun from Earth = x 10 8 km Radius of the Sun = 6.96 x 10 5 km What is the angular diameter subtended by the Sun and moon? What is the solid angle subtended by the Sun and moon? Which appears larger from the Earth?

8 Solid angle problem Geometric framework for calculating the solid angle subtended by a sphere of radius R whose center is a distance D from the observer

9 Solid angle problem (a) θ max = 2 sin -1 (R/D); Moon = 0.52º Sun = 0.53º (b) Solid angle of a cone with apex angle 2θ: 2π 0 θ 0 sinθ dθ dφ = 2π [ ] 0 θ = 2π cosθ θ 0 sinθ dθ = 2π (1 cosθ) Moon: Sun: (c) Sun subtends a solid angle 5% larger than the moon If these values were constant, could total solar eclipses be explained?

10 Formal definition of intensity Flux (F; measured on a surface normal to the beam) per unit solid angle (ω) traveling in a particular direction ˆ Ω I ˆ Ω ( ) = δf δω Typical units: Watts per square meter per steradian (W m -2 sr -1 ) Conservation of intensity: intensity (radiance) does not decrease with distance from the source (within a vacuum or other transparent medium) Contrast with flux density

11 Inverse square law Flux density at surface of sphere F F F F flux Irradiance (or flux density) decreases as the square of distance from the source Radiance is invariant with distance (note dependence of solid angle on r 2 ) Solar constant (irradiance at top of Earth s atmosphere) is ~1370 W m -2

12 Relationship between flux and intensity Flux (F): total power incident on a unit surface area Intensity (I): Flux contribution arriving from small element of solid angle along a direction ˆ Ω Hence, flux incident on or emerging from an arbitrary surface is found by integrating I over all relevant solid angles Since intensity is defined as the flux per unit solid angle normal to the beam, the contributions to the flux must be weighted by a factor of cos θ The flux density of radiation carried by a beam in the direction Ω through a surface element da is proportional to cos θ = ˆ n Ω ˆ

13 Relationship between flux and intensity The upward-directed flux from a surface is therefore given by: 2π 0 F = I cosθ dω = 2π π / I (θ,φ)cosθ sinθ dθ dφ What is the flux density of isotropic radiation? (i.e., if intensity is constant) F = 2π π / I (θ,φ)cosθ sinθ dθ dφ = 2π I cosθ sinθ dθ = 2π I = πi π / 2 0 π / sin 2θ dθ e.g., illumination of a horizontal surface under heavily overcast skies

14 Global insolation How much total solar radiation Φ is incident on Earth s atmosphere? Consider the amount of radiation intercepted by the Earth s disk 1370 W m -2 Φ = S 0 πr E 2 = W Applies for mean Sun-Earth distance of x 10 8 km But Earth s orbit is elliptical, so the solar flux (S) actually varies from 1330 W m -2 in July to 1420 W m -2 in January

15 Global insolation In addition to Earth s orbit, the power output from the Sun also varies over time (e.g., sunspot cycles) Furthermore, radiation is not uniformly incident on the surface but varies with incidence angle of the Sun Where is the peak? Houghton Note that this shows the amount of radiation incident at the top of the atmosphere only, not that which is available for absorption by the surface and atmosphere

16 Radiative properties of natural surfaces Illumination of the atmosphere from below by upwelling radiation

17 The fate of incident radiation Reflectivity, r λ (Albedo) Absorptivity, a λ Transmissivity, t λ (Transmittance) a λ + r λ + t λ = 1 Radiation incident on a surface is either reflected, absorbed or transmitted Conservation of energy dictates that the sum of these must be 100% (or 1) For opaque objects negligible transmission a λ + r λ = 1 and a λ = 1 - r λ If the object is highly reflective, a λ must be low (e.g., aluminum foil) Both a λ and r λ depend on the direction of the incident radiation

18 Transmission - Beer s Law The rate of power attenuation per unit distance is given by the absorption coefficient β a (with dimensions of inverse length), related to n i and wavelength λ by: β a = 4πn i λ For an initial intensity I 0 at position x = 0 in a medium, propagating in the x-direction: I(x) = I 0 e β a x I(x) I 0 = e βa x = t(x) Where t is the transmittance: the fraction of radiation that survives the trip over the distance x (i.e., that which is not absorbed) This equation is a form of a very important relationship known as BEER S LAW (also referred to as the Beer-Bouguer-Lambert Law)

19 Interaction of radiation with matter Reflection and refraction Absorption Note very small n i in the visible band, increasing in the UV and IR bands

20 Reflectivity R p = cosθ t m cosθ i cosθ t + m cosθ i R s = cosθ i m cosθ t cosθ i + m cosθ t 2 2 Fresnel relations for normal incidence: R normal = m 1 m + 1 (m is the relative refractive index) 2

21 Refractive index of aerosols Reflection and refraction Absorption

22 Absorption by silicates

23 Geometric optics When particle size is larger than wavelength, geometric optics or ray tracing can be used to analyze the interaction of EM waves with matter e.g., scattering of visible light by cloud ice particles (>50 µm) and raindrops (100 µm < r < 3 mm) Reflection and refraction at each boundary Sundog

24 Geometric optics Ray tracing predicts the geometry of the primary and secondary rainbows

25 Reflectivity spectra Which surface type has the generally highest albedo? Where is the visible reflectivity peak for grass? What about a dead plant vs. a live plant? What happens in the Fall? Note generally low reflectivity of vegetation, soil, and water, and strong contrast in reflectivity between visible and near-ir bands

26 MODIS images False Color: RR (Band 7: 1: 2155 True Color: (Band 670nm), nm),gg(band (Band2:4: nm), nm),bb(band (Band1:3: nm) nm)

27 Snow and cloud reflectance From:

28 Dirty snow/ice

29 MODerate resolution Imaging Spectroradiometer (MODIS) MODIS: a broadband, multispectral satellite instrument Flying on two polar-orbiting NASA satellites: Terra (10:30 am overpass) and Aqua (1:30 pm overpass) Swath width: 2330 km 36 spectral bands from VIS to IR, but not contiguous High radiometric resolution: 12-bit (0-4095) Spatial resolution: 250 m (bands 1-2), 500 m (bands 3-7), 1000 m (bands 8-36) Why the difference in spatial resolution between bands?

30 MODIS bands

31 MODIS spectral resolution Often specified by the full width at half-maximum (FWHM) response bandwidth > 100 nm FWHM = 100 nm in this case Jensen (2007), Remote sensing: an earth resources perspective, Prentice Hall.

32 Terra ASTER 14 bands Multispectral sensors Landsat TM 7 bands

33 Hyperspectral sensors NASA EO-1 satellite Launched in Nov 2000 Advanced Land Imager (ALI) Hyperion Atmospheric Corrector (AC)

35 Angular distribution of reflected radiation Specular: radiation incident at angle θ reflected away from surface at angle θ Lambertian: radiation incident at angle θ reflected equally at all angles Example of variable reflectance: glossy, semi gloss and flat paint

36 Sun glint MODIS visible imagery Feb 9, 2009 Observed in direction of reflection of the sun from smooth surfaces Specular reflection of visible light from smooth water surface

37 Lambertian reflectivity A Lambertian surface has the same radiance (or brightness) when viewed from any angle A common assumption in remote sensing using reflected radiation F r = rf i I = rf i π F r = π I r Flux density of isotropic radiation derived earlier I = rs 0 cosθ i π Assume incident flux is entirely due to direct solar radiation from a zenith angle θ i, where S 0 is the solar flux Note that this is an assumption that simplifies radiative transfer calculations the actual angular distribution of reflection must be determined from measurements

38 Albedo and solar heating Surface albedo determines direct heating by sunlight and hence the heating of adjacent air Compare bare soil (albedo ~10%) with fresh, dry snow (albedo ~90%) Surface type therefore affects the temperature of the overlying air

39 Satellite imaging in the Visible and Near-IR Schematic viewing geometry for satellite remote sensing measurements

40 Visible image from GOES-East, 3 Feb 2010 What time was this image acquired?

41 Geostationary Satellite Imagery from UW-Madison

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