1. Basic introduction to electromagnetic field. wave properties and particulate properties.

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1 Lecture Baic Radiometric Quantitie. The Beer-Bouguer-Lambert law. Concept of extinction cattering plu aborption and emiion. Schwarzchild equation. Objective:. Baic introduction to electromagnetic field: Definition, dual nature of electromagnetic radiation, electromagnetic pectrum.. Baic radiometric quantitie: energy, intenity, and flux. 3. The Beer-Bouguer-Lambert law. Concept of extinction cattering + aborption and emiion. Optical depth. 4. Simple apect of radiative tranfer: Schwarzchild radiative tranfer equation. Required reading: L:.,.4. Baic introduction to electromagnetic field. Electromagnetic radiation i a form of tranmitted energy. Electromagnetic radiation i o-named becaue it ha electric and magnetic field that imultaneouly ocillate in plane mutually perpendicular to each other and to the direction of propagation through pace. Electromagnetic radiation exhibit the dual nature: wave propertie and particulate propertie. Wave nature of radiation: radiation can be thought of a a traveling wave. Electromagnetic wave are characterized by wavelength or frequency and peed. The peed of light in a vacuum: c.9979x 8 m/ 3.x 8 m/

2 Wavelength,,, i the ditance between two conecutive peak or trough in a wave. Frequency, ν ~, i defined a the number of wave cycle per econd that pa a given point in pace. Wavenumber, ν, i defined a a count of the number of wave cret or trough in a given unit of length. Relation between,, ν and ν ~ : ν ν ~ /c / [.] UNTS: Wavelength unit: LENGTH, Angtrom A : A x - m Nanometer nm: nmx -9 m Micrometer m: m x -6 m Frequency unit: unit cycle per econd / or - i called hertz abbreviated Hz Wavenumber unit: LENGTH - often in cm - NOTE: ν[ cm cm m ] [ m] EXAMPLE: 8- m atmopheric window i cm - Particulate nature of radiation: Radiation can be alo decribed in term of particle of energy, called photon. The energy of a photon i given by the expreion: E photon h ~ ν h c/ h cν [.], where h i Plank contant h 6.656x -34. NOTE: Plank contant h i very mall!

Eq. [.] relate energy of each photon of the radiation to the electromagnetic wave characteritic ν ~ and. Spectrum of electromagnetic radiation: Figure. The electromagnetic pectrum.

3 Eq. [.] relate energy of each photon of the radiation to the electromagnetic wave characteritic ν ~ and. Spectrum of electromagnetic radiation: Figure. The electromagnetic pectrum. n thi coure we tudy the UV, viible and infrared radiation. Table. Relationhip between radiation component tudied in thi coure. Name of Wavelength Spectral equivalence pectral region region, m Solar. - 4 Ultraviolet + Viible + Near infrared Shortwave Terretrial 4 - Far infrared Longwave nfrared.75 - Near infrared + Far infrared Ultraviolet Near ultraviolet + Far ultraviolet UV-A + UV-B + UV-C + Far ultraviolet Shortwave. - 4 Solar Near infrared + Viible + Ultraviolet Longwave 4 - Terretrial Far infrared Viible Shortwave - Near infrared - Ultraviolet Near infrared.75-4 Solar - Viible - Ultraviolet nfrared - Far infrared Far infrared 4 - Terretrial Longwave nfrared - Near infrared Thermal 4 - Terretrial Longwave Far infrared 3

4 . Baic radiometric quantitie. Solid angle i the angle ubtended at the center of a phere by an area on it urface numerically equal to the quare of the radiu UNTS: of a olid angle teradian r Ω r σ σ Ω r A differential olid angle can be expreed a d σ d Ω in θ d θdφ, r uing that a differential area i dσ r dθ r inθ dφ EXAMPLE: Solid angle of a unit phere 4π EXAMPLE: What i the olid angle of the Sun from the Earth if the ditance from the Sun from the Earth i d.5x 8 km and Sun radiu i R 6.96x 5 km. Ω π d R 6.76 x 5 r ntenity or radiance i defined a radiant energy in a given direction per unit time per unit wavelength or frequency range per unit olid angle per unit area perpendicular to the given direction: de co θ dωdtdad i referred to a monochromatic intenity. [.3] Monochromatic doe not mean at a ingle wavelength, but in a very narrow infiniteimal range of wavelength centered at. NOTE: ame name: intenity pecific intenity radiance UNTS: from Eq.[.3]: ec - r - m - m - W r - m - m - 4

5 Propertie of intenity: n general, intenity i a function of the coordinate r, direction Ω r, wavelength or frequency, and time. Thu, it depend on even independent variable: three in pace, two in angle, one in wavelength or frequency and one in time. ntenity, a a function of poition and direction, give a complete decription of the electromagnetic field. f intenity doe not depend on the direction, the electromagnetic field i aid to be iotropic. f intenity doe not depend on poition the field i aid to be homogeneou. Flux or irradiance i defined a radiant energy in a given direction per unit time per unit wavelength or frequency range per unit area perpendicular to the given direction: UNTS: from Eq.[.4]: ec - m - m - W m - m - de F [.4] dtdad From Eq. [.3]-[.4]: Ω F coθ dω [.5] Thu, monochromatic flux i the integration of normal component of monochromatic intenity over ome olid angle. Monochromatic upwelling upward hemipherical flux on a horizontal plane i the integration of normal component of monochromatic intenity over the all olid angle in the upper hemiphere. Eq. [.5] in pherical coordinate give: F ππ / θ, co θ in θ dθd π, d d where coθ 5

6 Monochromatic net flux i the integration of normal component of monochromatic intenity over the all olid angle over 4π. Net flux for a horizontal plane i the difference in upwelling and downwelling hemipherical fluxe: F net, F F π, d d ntegral or total intenity and flux F are determined by integrating over the wavelength the monochromatic intenity and flux, repectively: d F F d ntenitie and fluxe may be per wavelength or per wavenumber. Since intenity acro a pectral interval mut be the ame, we have d dν and thu ν d ν dν ν EXAMPLE: Convert between radiance in per wavelength to radiance per wavenumber unit at m. Given 9.9 W m - r - m -. What i ν? ν 9.9 W m - r - m - m -3 cm.99 W m - r - cm The Beer-Bouguer-Lambert law. Concept of extinction cattering + aborption and emiion. Extinction and emiion are two main type of the interaction between an electromagnetic radiation field and a medium e.g., the atmophere. General definition: Extinction i a proce that decreae the radiant intenity, while emiion increae it. NOTE: ame name : extinction attenuation 6

7 Radiation i emitted by all bodie that have a temperature above abolute zero O K often referred to a thermal emiion. Extinction i due to aborption and cattering. Aborption i a proce that remove the radiant energy from an electromagnetic field and tranfer it to other form of energy. Scattering i a proce that doe not remove energy from the radiation field, but may redirect it. NOTE: Scattering can be thought of a aborption of radiant energy followed by reemiion back to the electromagnetic field with negligible converion of energy. Thu, cattering can remove radiant energy of a light beam traveling in one direction, but can be a ource of radiant energy for the light beam traveling in other direction. The fundamental law of extinction i the Beer-Bouguer-Lambert law, which tate that the extinction proce i linear in the intenity of radiation and amount of matter, provided that the phyical tate i.e., T, P, compoition i held contant. NOTE: Some non-linear procee do occur a will be dicued later in the coure. Conider a mall volume V of infiniteimal length d and area A containing optically active matter. Thu, the change of intenity along a path d i proportional to the amount of matter in the path. For extinction: For emiion: d e d, [.6] d e, d [.7] where e, i the volume extinction coefficient LENGTH - and i the ource, function. + d d 7

8 n the mot general cae, the ource function ha emiion and cattering contribution. Generally, the volume extinction coefficient i a function of poition. Sometime it may be expreed mathematically a e,, but i often dropped. NOTE: Volume extinction coefficient i often referred to a the extinction coefficient., Extinction coefficient aborption coefficient + cattering coefficient e, a, +, [.8] NOTE: Extinction coefficient a well a aborption and cattering coefficient can be expreed in different form according to the definition of the amount of matter e.g., number concentration, ma concentration, etc. of matter in the path ee Lecture 4. Volume and ma extinction coefficient are mot often ued. Ma extinction coefficient volume extinction coefficient/denity UNTS: the ma coefficient i in unit area per unit ma LENGTH MASS -. For intance: cm g -, m kg -, etc. f ρ i the denity ma concentration of a given type of particle or molecule, then * e, ρ e, ρ *,, [.9] a, ρk where the * e, *,, and k are the ma extinction, cattering, and aborption coefficient, repectively. NOTE: L ue k for both ma extinction and ma aborption coefficient! 8

9 The extinction cro ection of a given particle or molecule i a parameter that meaure the attenuation of electromagnetic radiation by thi particle or molecule. n the ame fahion, cattering and aborption cro ection can be defined. UNTS: the cro ection i in unit area LENGTH f N i the particle or molecule number concentration of a given type of particle or molecule, then where σ e,, σ,,, e, σ e, σ, N N, [.] a, σ a,, and σ a, are the extinction, cattering, and aborbing cro ection, repectively. UNTS: Particle number concentration i in the number of particle per unit volume LENGTH -3. N Optical depth of a medium between point and i defined a e, d UNTS: optical depth i unitle. S S NOTE: ame name : optical depth optical thickne optical path f e, doe not depend on poition called a homogeneou optical path, thu e,,, < e, > and < e, > < e, > For thi cae, the Extiction law can be expreed a exp exp < e, > [.] Optical depth can be expreed in everal way: 9

10 * e, d ρe, d Nσ e, d [.] f in a given volume there are everal type of optically active particle each with i e,, etc., then the optical depth can be expreed a: i * i e, d ρie, d i Niσ e, d [.3] i i where ρ i and N i i the ma concentration denitie and particle concentration of the i-th pecie. i 4. Simple apect of radiative tranfer. Let conider a mall volume V of infiniteimal length d and area A containing optically active matter. Uing the Extinction law, the change lo plu gain due to both the thermal emiion and cattering of intenity along a path d i d Dividing thi equation by e, d, we find, e, d+ e, e, d d + d [.4] Eq. [.4] i the differential equation of radiative tranfer called Schwarzchild equation. NOTE: Both and are generally function of both poition and direction. The optical depth i Thu, e d d e d, '

11 Uing the above expreion for d, we can re-write Eq. [.4] a d d d d + or a [.5] Thee are other form of the differential equation of radiative tranfer. Let re-arrange term in the above equation and multiply both ide by exp-. We have exp d d + exp exp and uing that d[xexp-x]exp-xdx-exp-xxdx we find [ exp ] exp d d Then integrating over the path from to, we have and [ exp ] exp d d [ exp ] exp d Thu exp exp d and, uing d e d, we have a olution of the equation of radiative, tranfer often referred to a the integral form of the radiative tranfer equation:

12 exp + exp e, d [.6] NOTE: i The above equation give monochromatic intenity at a given point propagating in a given direction often called an elementary olution. A completely general ditribution of intenity in angle and wavelength or frequencie can be obtained by repeating the elementary olution for all incident beam and for all wavelength or frequencie. ii Knowledge of the ource function i required to olve the above equation. n the general cae, the ource function conit of thermal emiion and cattering or emiion from cattering, depend on the poition and direction, and i very complex. One may ay that the radiative tranfer equation i all about the ource function. Plane-parallel atmophere. For many application, the atmophere can be approximated by a plane-parallel model to handle the vertical tratification of the atmophere. Plane-parallel atmophere conit of a certain number of atmopheric layer each characterized by homogeneou propertie e.g., T, P, optical propertie of a given pecie, etc. and bordered by the bottom and top infinite plate called boundarie. Traditionally, the vertical coordinate z i ued to meaure linear ditance in the plane-parallel atmophere: z coθ where θ denote the angle between the upward normal and the direction of propagation of a light beam or zenith angle and φ i the azimuthal angle. x r z r z φ θ r r y

13 Uing d dz/coθ, the radiative tranfer equation can be written a d z θ co θ dz e, z θ + z θ ntroducing the optical depth meaured from the outer boundary downward a z z z, z e z dz and uing d e z dz and coθ, we have, d [.7] d Eq. [.7] i the baic equation for the problem of radiative tranfer in the plane-parallel atmophere Eq.[.7] may be olved to give the upward or upwelling and downward or downwelling intenitie for a finite atmophere which i bounded on two ite. Upward intenity i for or θ π / Downward intenity i for or π / θ π uing that co coπ/ and coπ - 3

14 4 Plane-parallel atmophere. NOTE: For downward intenity, i replaced by. The radiative tranfer equation [.7] can be written for upward and downward intenitie: d d [.8a] d d [.8b] A olution of Eq.[.8a] give a upward intenity in the plane-parallel atmophere: + d exp exp [.9a] z z z z Bottom Top

15 5 and a olution of Eq.[.8b] give a downward intenity in the plane-parallel atmophere: + d exp exp [.9b]

Lecture 3 Basic radiometric quantities.

Lecture 3 Basic radiometric quantities. Lecture 3 Baic radiometric quantitie. The Beer-Bouguer-Lambert law. Concept of extinction cattering plu aborption and emiion. Schwarzchild equation.. Baic introduction to electromagnetic field: Definition,