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Section 3. Radiative Transfer

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Section 3. Radiative Transfer Section 3. Radiative Transfer References • Kidder and Vonder Haar: chapter 3 • Stephens: chapter 2, pp. 65-76; chapter 3, pp. 81-87, 99-121 • Liou: chapter 1; chapter 2, pp. 38-41; chapter 3, pp. 53-56, 60-63; chapter 4, pp. 87-93 • Lenoble: parts of chapters 1,3,5,6,17 "All of the information received by a satellite about the Earth and its atmosphere comes in the form of electromagnetic radiation." Therefore, we need to know • how this radiation is generated • how it interacts with the atmosphere 3.1 Electromagnetic Radiation Electromagnetic (EM) waves are generated by oscillating electric charges which generate an oscillating electric field. This produces a magnetic field, which further produces an electric field. These fields thus propagate outwards from the charge, creating each other. E ⊥ H ⊥ direction of propagation. ELECTRIC FIELD E λ DIRECTION OF PROPAGATION MAGNETIC FIELD H EM radiation is usually specified by • wavelength (λ) = distance between crests • frequency (ν) = number of oscillations per second = c/λ • wavenumber ( ν or κ) = number of crests per unit length (usually cm-1) = 1/λ ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-1 Can usually approximate the speed of light in the atmosphere by that in a vacuum. However, the change in density and humidity with height can cause significant refraction (bending of the EM rays) which must be taken into account in satellite pointing. The EM spectrum can be divided into different spectral regions, which are useful in remote sounding: • ultraviolet (UV) ~ 100 to 400 nm • visible ~ 400 to 700 nm • infrared ~ 700 nm to 1 mm • microwave ~ 1 mm to 1 m • radio ~ includes microwave to > 100 m See figure (2.3) - the electromagnetic spectrum. EM radiation carries energy that can be detected by sensors. The energy per unit area per unit time flowing perpendicularly into a surface is given by the Poynting vector S : S = c 2ε E × H units of W/m2 o ENERGY S (W / m 2 ) where ε = vacuum permittivity o E = electric field H = magnetic field A = 1 m 2 E , H, and S oscillate rapidly, so S is difficult to measure instantaneously. Usually measure the average magnitude over some time interval: F = S = radiant flux density (W/m2) The radiant flux density is redefined based on the direction of energy travel: radiant exitance (M) = radiant flux density emerging from an area irradiance (E) = radiant flux density incident on an area RADIANT EXITANCE, M IRRADIANCE, E ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-2 Radiation is also a function of direction. This is accounted for by using the solid angle. solid angle (Ω ≠ right ascension!) the area of the projection onto a unit sphere of an object, where lines are drawn from the centre of the sphere to every point on the surface of the object z OBJECT (AREA A) r PROJECTION ONTO UNIT SPHERE = SOLID ANGLE Ω y r = 1 x UNIT SPHERE π AREA = 4 SPHERE OF RADIUS r AREA = 4 π r Examples: • solid angle of an object that completely surrounds a point: Ω = 4π steradians (sr) • solid angle of an infinite plane: Ω = 2π sr • solid angle of an object of cross sectional area A at distance r from a point: Ω = A / r 2 Ω A dA (using fractional areas = or dΩ = ) 4π 4πr 2 r 2 Differential element of solid angle is given by: dΩ = sin θ dθ dφ = dµ dϕ where µ = cos θ θ = zenith angle = 90° – elevation angle φ = azimuth angle See figure (K&VH 3.4) - mathematical representation of solid angle. We can now define radiance or intensity (I or L) = radiant flux density per unit solid angle S I = W m-2 sr-1 dΩ ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-3 This is the most important of the radiometric terms that we have defined. Strictly, the radiance represents the EM radiation leaving or incident upon an area perpendicular to the beam. For other directions, it must be weighted by cos θ. What is the radiant exitance, M, (i.e., the total amount of radiation leaving the surface) for a small surface area emitting radiance I? 2π 2ππ/ 2 2π 1 M = ∫I(θ,ϕ)cosθdΩ = ∫∫I(θ,ϕ)cosθsinθdθdϕ = ∫∫I(µ,ϕ)µdµdϕ 0 0 0 0 0 i.e., The radiant exitance M is obtained by integrating the radiance I over a hemisphere 2ππ/ 2 For I independent of direction (isotropic): M = I∫∫cos θ sin θ dθ dϕ = π I. 0 0 The radiation may be wavelength dependent. → prefix the energy-dependent terms by "monochromatic" or "spectral". Symbols are subscripted accordingly, e.g., Iλ, Iν, Iν refers to I per unit λ, ν, ν. Units: e.g. monochromatic radiance: Iλ in W m-2 sr-1 nm-1 Iν in W m-2 sr-1 Hz-1 -2 -1 -1 -1 Iν in W m sr (cm ) The total radiance is then the integral over λ, ν, or ν of the monochromatic radiance. ∞ ∞ ∞ = λ = ν = ν I ∫ Iλ d ∫ Iν d ∫ Iν d 0 0 0 ν ν 2 d 2 Iλ = − Iν = Iν = ν Iν so dλ c λ Iλ = ν Iν = ν Iν i.e., The radiation per unit λ interval is the same as radiation per unit ν or ν interval, if the sizes of the intervals are accounted for. Radiance is a useful quantity for satellite measurements because it is independent of distance from an object as long as the viewing angle and the amount of intervening matter do not change. (Both the irradiance and the solid angle decrease with r2 so I remains constant.) See table (K&VH 3.1) - radiation symbols and units. ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-4 3.2 Blackbody Radiation All matter emits radiation if it is at a temperature > absolute zero. A blackbody is a perfect emitter - it emits the maximum possible amount of radiation at each wavelength. A blackbody is also a perfect absorber, absorbing at all wavelengths of radiation incident on it. Therefore, it looks black. Planck's Blackbody Function No real materials are perfect blackbodies. However, the radiation inside a cavity (whose walls are opaque to all radiation) is the radiation that would be emitted by a hypothetical blackbody at the same temperature. The cavity walls emit, absorb, and reflect radiation until equilibrium is reached. Planck postulated that atoms oscillating in the walls of the cavity have discrete energies given by E = n h ν where n = integer (quantum number) h = Planck's constant ν = frequency A quantum of energy emitted when an atom changes its energy state is then ∆E = h ν (∆n = 1). Using these two assumptions, Planck derived the blackbody function, describing the radiance emitted by a blackbody 2hc 2λ−5 Bλ (T) = hc exp − 1 λkT where -2 -1 -1 Bλ = monochromatic radiance (W m sr µm ) k = Boltzmann's constant T = absolute temperature λ−5 c1 This can be written as: Bλ (T) = c exp 2 −1 λT where -16 -2 -1 c1 = first radiation constant (1.191 × 10 W m sr ) -2 c2 = second radiation constant (1.439 × 10 m K) ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-5 ∴ The radiance emitted by a blackbody depends only on λ and T. • Bλ(T) increases with temperature • the λ of maximum Bλ(T) decreases with temperature B Sketch B vs. λ curves for: T1 > T2 > T3 > T4 λ Several other relations can be derived from Planck's blackbody function. (1) Wien's Displacement Law ∂B λ (T) The wavelength at which Bλ(T) is a maximum is determined using = 0 . ∂λ 2897.9 This gives λ = µm, for T in K → this is Wien's displacement Law. m T The hotter the object, the shorter the λ of its maximum intensity (e.g., element on a stove). This law can be used to determine the T of a blackbody from the position of the maximum monochromatic radiance. (2) Stefan Boltzmann Law The monochromatic radiant exitance is simply Mλ(T) = π Bλ(T) because the blackbody radiance is isotropic (independent of direction). The total radiant exitance from a blackbody is then ∞∞ π5 c M(T) = M (T)dλ = πB (T)dλ = ... = 1 T 4 ∫∫λ λ 4 00 15 c 2 σT 4 ∴ M (T) = σT 4 or B(T) = → this is the Stefan Boltzmann Law. BB π where σ = Stefan Boltzmann constant = 5.67 × 10-8 W m-2 K-4 This states that the total amount of radiation emitted from a given surface area is proportional to T4. ___________________________________________________________________________________ PHY 499S – Earth Observations from Space, Spring Term 2005 (K. Strong) page 3-6 (3) Rayleigh Jeans Approximation At longer wavelengths, the Planck blackbody function can be simplified. For λ in the microwave region, c2 / λT << 1 for T relevant to Earth. c c c T ∴ exp 2 ≈ 1 + 2 , giving B (T) ≈ 1 λ λ λ λ4 T T c 2 → this is the Rayleigh Jeans Approximation. It states that in the microwave region, the radiance is simply linearly proportional to T. -4 Often the radiance will be scaled by (c1/c2)λ to get the brightness temperature, Tb, which is radiance expressed in units of temperature. Tb is the temperature required to match the measured intensity to the Planck blackbody function at a given λ. Tb is also used in the infrared (referred to as the equivalent brightness temperature), but it must be derived from the blackbody function.
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