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Material Type: Notes; Class: Special Topics; Subject: Earth and Atmospheric Sciences; University: Georgia Institute of Technology-Main Campus; Term: Unknown 2005;

Typology: Study notes

Pre 2010

1 / 15

Download Basic Radiometric Quantities. The Beer-Bouguer-Lambert Law | EAS 8803 and more Study notes Environmental Science in PDF only on Docsity! 1 Lecture 3 Basic radiometric quantities. The Beer-Bouguer-Lambert law. Concepts of extinction (scattering plus absorption) and emission. Schwarzschild’s equation. Objectives: 1. Basic introduction to electromagnetic field: Definitions, dual nature of electromagnetic radiation, electromagnetic spectrum. 2. Basic radiometric quantities: energy, intensity, and flux. 3. The Beer-Bouguer-Lambert law. Concepts of extinction (scattering + absorption) and emission. Optical depth. 4. Simple aspects of radiative transfer: Schwarzschild’s radiative transfer equation. Required reading: L02: 1.1, 1.4 1. Basic introduction to electromagnetic field. Electromagnetic radiation is a form of transmitted energy. Electromagnetic radiation is so-named because it has electric and magnetic fields that simultaneously oscillate in planes mutually perpendicular to each other and to the direction of propagation through space. Electromagnetic radiation exhibits the dual nature: it has wave properties and particulate properties. Figure 3.1 Electromagnetic radiation as a traveling wave. 2 Wave nature of radiation: radiation can be thought of as a traveling wave characterized by the wavelength (or frequency, or wavenumber) and speed. NOTE: speed of light in a vacuum: c = 2.9979x108 m/s ≅ 3.00x108 m/s. Wavelength, λ, is the distance between two consecutive peaks or troughs in a wave. Frequency, ν~ , is defined as the number of waves (cycles) per second that pass a given point in space. Wavenumber, ν , is defined as a count of the number of wave crests (or troughs) in a given unit of length. Relation between λ, ν and ν~ : ν = ν~ /c = 1/λ [3.1] NOTE: The frequency is a more fundamental quantity than the wavelength Wavelength units: LENGTH, Angstrom (A): 1 A = 1x10-10 m; Nanometer (nm): 1 nm=1x10-9 m; Micrometer (µm): 1 µm = 1x10-6 m; Frequency units: unit cycles per second 1/s (or s-1) is called Hertz (abbreviated Hz) Wavenumber units: LENGTH-1 (often in cm-1) • As a transverse wave, EM radiation can be polarized. Polarization is the distribution of the electric field in the plane normal to propagation direction. Particulate nature of radiation: Radiation can be also described in terms of particles of energy, called photons. The energy of a photon is given by the expression: E photon = h ν~ = h c/λ = h cν [3.2] where h is Plank’s constant (h = 6.6256x10-34 J s). NOTE: Plank’s constant h is very small! • Eq. [3.2] relates energy of each photon of the radiation to the electromagnetic wave characteristics (ν~ , ν or λ). 5 Solid angle is the angle subtended at the center of a sphere by an area on its surface numerically equal to the square of the radius 2r σ =Ω UNITS: of a solid angle = steradian (sr) EXAMPLE: Solid angle of a unit sphere = 4π EXAMPLE: What is the solid angle of the Sun from the Earth if the distance from the Sun from the Earth is d=1.5x108 km and Sun’s radius is Rs = 6.96x105 km. srx d Rs 5 2 2 1076.6 −==Ω π Properties of intensity: In general, intensity is a function of the coordinates ( rr ), direction ( Ω r ), wavelength (or frequency), and time. Thus, it depends on seven independent variables: three in space, two in angle, one in wavelength (or frequency) and one in time. Intensity, as a function of position and direction, gives a complete description of the electromagnetic field. If intensity does not depend on the direction, the electromagnetic field is said to be isotropic. If intensity does not depend on position the field is said to be homogeneous. r Ω σ A differential solid angle can be expressed as φθθσ dd r dd )sin(2 ==Ω , using that a differential area is dΑ = (r dθ) (r sin(θ) dφ) 6 Flux (or irradiance) is defined as radiant energy in a given direction per unit time per unit wavelength (or frequency) range per unit area perpendicular to the given direction: λ λ λ dtdAd dEF = [3.4] UNITS: from Eq.[3.4]: (J sec-1 m-2 µm-1) = (W m-2 µm-1) From Eqs. [3.3]-[3.4]: ∫ Ω Ω= dIF )cos(θλλ [3.5] Thus, monochromatic flux is the integration of normal component of monochromatic intensity over some solid angle. Monochromatic upwelling (upward) hemispherical flux on a horizontal plane is the integration of normal component of monochromatic intensity over the all solid angles in the upper hemisphere ϕµµϕµϕθθθϕθ λ ππ π λλ ddIddIF ∫∫∫ ∫ ↑↑↑ == 1 0 2 0 2 0 2/ 0 ),()sin()cos(),( [3.7] where µ = cos(θ). Downwelling (downward) hemispherical flux (i.e., integration over the lower hemisphere) =−=−= ∫∫∫ ∫ − ↓↓↓ ϕµµϕµϕθθθϕθ λ ππ π π λλ ddIddIF 1 0 2 0 2 0 2/ ),()sin()cos(),( ϕµµϕµλ π ddI∫∫ −= ↓ 1 0 2 0 ),( [3.8] ∫ ΩΩ= ↑↑ π λλ 2 )( dnIF r r [3.6] Eq. [3.6] in spherical coordinates gives 7 Monochromatic net flux is the integration of normal component of monochromatic intensity over the all solid angles (over 4π). Net flux for a horizontal plane is the difference in upwelling and downwelling hemispherical fluxes: ϕµµϕµλ π λλλ ddIFFFnet ∫∫ − ↓↑ =−= 1 1 2 0 ),(, [3.9] Actinic flux is the total spectral energy at point (used in photochemistry): ∫ ΩΩ= π λλ 4 , )( dIFact [3.10] Spectral integration: Radiative quantities may be spectrally integrated. For example, the downwelling shortwave flux is ∫ ↓↓ = m m dFF µ µ λ λ 0.4 1.0 [3.11] Intensities and fluxes may be per wavelength or per wavenumber. Since intensity across a spectral interval must be the same, we have νλ νλ dIdI = and thus 22 1 λ νν λ λλλν IId dII === [3.12] EXAMPLE: Convert between radiance in per wavelength to radiance per wavenumber units at λ = 10 µm. Given Iλ = 9.9 W m-2 sr-1µm-1. What is Iν ? Iν = (9.9 W m-2 sr-1µm-1) (10 µm) (10-3 cm) = 0.099 W m-2 sr-1(cm-1)-1 10 The extinction cross section of a given particle (or molecule) is a parameter that measures the attenuation of electromagnetic radiation by this particle (or molecule). In the same fashion, scattering and absorption cross sections can be defined. UNITS: the cross section is in unit area (LENGTH2) If N is the particle (or molecule) number concentration of a given type of particles (or molecules), then Nee λλ σβ ,, = Nss λλ σβ ,, = [3.17] Naa λλ σβ ,, = where σe,λ, σs,λ , and σa,λ are the extinction, scattering, and absorbing cross sections, respectively. UNITS: Particle number concentration is in the number of particles per unit volume (LENGTH-3). Optical depth of a medium between points s1 and s2 is defined as dssss s s e )();( 2 1 ,12 ∫= λλ βτ UNITS: optical depth is unitless. NOTE: “same name”: optical depth = optical thickness = optical path • If βe,λ (s) does not depend on position (called a homogeneous optical path), thus βe,λ (s) = <βe,λ > and sssss ee >=<−>=< λλλ ββτ ,12,12 )();( For this case, the Extiction law can be expressed as )exp()exp( ,00 sIII e ><−=−= λλ βτ [3.18] s S1 S2 τλ 11 Optical depth can be expressed in several ways: ∫∫∫ === 2 1 2 1 2 1 , * ,,21 );( s s e s s e s s e dsNdsdsss λλλλ σρββτ [3.19] • If in a given volume there are several types of optically active particles each with βie,λ , etc., then the optical depth can be expressed as: ∫∑∫∑∫∑ === 2 1 2 1 2 1 , * ,, s s i ei i s s i ei i s s i e i dsNdsds λλλλ σβρβτ [3.20] where ρi and Ni is the mass concentrations (densities) and particles concentrations of the i-th species. 4. Simple aspects of radiative transfer. Let’s consider a small volume ∆V of infinitesimal length ds and area ∆A containing optically active matter. Using the Extinction law, the change (loss plus gain due to both the thermal emission and scattering) of intensity along a path ds is dsJdsIdI ee λλλλλ ββ ,, +−= Dividing this equation by βe,λ ds, we find λλ λ λ β JI ds dI e +−= , [3.21] Eq. [3.21] is the differential equation of radiative transfer called Schwarzchild’s equation. NOTE: Both Iλ and Jλ are generally functions of both position and direction. The optical depth is dssss s s e )();( 1 ,1 ∫= λλ βτ Thus dssd e )(,λλ βτ −= Using the above expression for dτλ, we can re-write Eq. [3.21] as s' 0 s1 s” τλ(s1;s’) τλ(s1;s”) 12 λλ λ λ τ JI d dI +−=− or as [3.22] λλ λ λ τ JI d dI −= These are other forms of the differential equation of radiative transfer. Let’s re-arrange terms in the above equation and multiply both sides by exp(-τλ). We have λλλλ λ λλ ττ τ τ JI d dI )exp()exp()exp( −=−+−− and (using that d[I(x)exp(-x)]=exp(-x)dI(x)-exp(-x)I(x)dx) we find [ ] λλλλλ τττ dJId )exp()exp( −=−− Then integrating over the path from 0 to s1 , we have [ ] λλλλλ τττ dJsssssId s s ));(exp());(exp()( 1 0 0 1 1 1 ∫ ∫ −=−− and [ ] λλλλλλ τττ dJsssIsI s ∫ −=−−− 1 0 111 ));(exp())0;(exp()0()( Thus λλλλλλ τττ dJsssIsI s ∫ −−−= 1 0 111 ));(exp())0;(exp()0()( and, using dssd e )(,λλ βτ −= , we have a solution of the equation of radiative transfer (often referred to as the integral form of the radiative transfer equation): dsJsssIsI e s λλλλλλ βττ , 0 111 1 ));(exp())0;(exp()0()( ∫ −+−= [3.23]