Lecture 3 Basic radiometric quantities.

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Lecture 3 Baic radiometric quantitie. The Beer-Bouguer-Lambert law. Concept of extinction cattering plu aborption and emiion. Schwarzchild equation.

1 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, 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 Differential and integral form of the 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: it ha wave propertie and particulate propertie. igure 3. Schematic repreentation of electromagnetic radiation a a traveling wave.

2 Wave nature of radiation: radiation can be thought of a a traveling wave characterized by the wavelength or frequency, or wavenumber and peed. NOTE: peed of light in a vacuum: c =.9979x 8 m/ 3.x 8 m/. Wavelength, i the ditance between two conecutive peak or trough in a wave. requency, ~, i defined a the number of wave cycle per econd that pa a given point in pace. Wavenumber,, i defined a a count of the number of wave cret or trough in a given unit of length. Relation between and ~ = ~ c = / NOTE: The frequency i a more fundamental quantity than the wavelength Wavelength unit: LENGTH, Angtrom A: A = x - m Nanometer nm: nm=x -9 m Micrometer m: m = x -6 m requency unit: unit cycle per econd / or - i called Hertz abbreviated Hz Wavenumber unit: LENGTH - often in cm - A a tranvere wav EM radiation can be polarized. Polarization i the ditribution of the electric field in the plane normal to propagation direction. Particulate nature of radiation: Radiation can be alo decribed in term of particle of energy, called photon. The energy of a photon i: E photon = h ~ = h c=h c where h i Plank contant h = 6.656x -34. NOTE: Plank contant h i very mall! Eq. [3.] relate energy of each photon of the radiation to the electromagnetic wave characteritic ~, or

Spectrum of electromagnetic radiation i the ditribution of electromagnetic radiation according to energy or, equivalently, according to the wavelength, wavenumber, or frequency. igure 3.

- 4 Ultraviolet + Viible + Near infrared = Shortwave Terretrial 4 - ar infrared = Longwave nfrared.75 - Near infrared + ar infrared Ultraviolet. -.38 Near ultraviolet + ar ultraviolet = UV-A + UV-B + UV-C + ar ultraviolet Shortwave.

3 Spectrum of electromagnetic radiation i the ditribution of electromagnetic radiation according to energy or, equivalently, according to the wavelength, wavenumber, or frequency. igure 3. The electromagnetic pectrum. Table 3. 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 - ar infrared = Longwave nfrared.75 - Near infrared + ar infrared Ultraviolet Near ultraviolet + ar ultraviolet = UV-A + UV-B + UV-C + ar ultraviolet Shortwave. - 4 Solar = Near infrared + Viible + Ultraviolet Longwave 4 - Terretrial = ar infrared Viible Shortwave - Near infrared - Ultraviolet Near infrared.75-4 Solar - Viible - Ultraviolet = nfrared - ar infrared ar infrared 4 - Terretrial = Longwave = nfrared - Near infrared Thermal 4 - Terretrial = Longwave = ar infrared 3

4 . Baic radiometric quantitie. lux and intenity are the two meaure of the trength of an electromagnetic field that are central to mot problem in radiative tranfer cience. 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 ee figure 3.3: d de co ddtdad i called the monochromatic intenity. Monochromatic doe not mean at a ingle wavelength, but in a very narrow infiniteimal range of wavelength d centered at. NOTE: ame name: intenity = pecific intenity = radiance [3.3] UNTS: from Eq.[3.3]: ec - r - m - m - = W r - m - m - igure 3.3 llutration of differential olid angle in pherical coordinate. 4

5 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 A r r A A differential olid angle can be expreed a da d dd r in, uing that a differential area i da = r d r in d EXAMPLE: Solid angle of a unit phere = 4 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. D R 6.76x 5 r Propertie of intenity: n general, intenity i a function of the coordinate r, direction, wavelength or frequency, and time. Thu, it depend on even independent variable: three in pac two in angl 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. 5

6 lux 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: de d [3.4] dtdad UNTS: from Eq.[3.4]: ec - m - m - = W m - m - rom Eq. [3.3]-[3.4]: co d [3.5] Monochromatic flux i the integration of normal component of monochromatic intenity over a certain olid angle. Monochromatic upwelling upward hemipherical flux on a horizontal plane i the integration of the normal component of monochromatic intenity over the olid angle of the hemiphere nd where co n [3.6] Eq. [3.6] in pherical coordinate give /, co in dd, dd [3.7] where = co. Downwelling downward hemipherical flux i.e., integration over the lower hemiphere /, co in dd, dd, dd [3.8] 6

7 Monochromatic net flux i the integration of normal component of monochromatic intenity over the entire olid angle over 4. Net flux for a horizontal plane i the difference in upwelling and downwelling hemipherical fluxe: net,, dd [3.9] Actinic flux i the total pectral energy at point ued in photochemitry: act d [3.], 4 Spectral integration: Radiative quantitie can be pectrally integrated e.g., in energy balance calculation: SW and LW. or exampl the downwelling hortwave SW flux i 4.m.m d [3.a] and the upwelling SW flux i 4.m.m d [3.b] Similarly, the downwelling and upwelling longwave LW fluxe are m 4m m 4m d d [3.c] [3.d] 7

8 Photoynthetically Active Radiation PAR deignate the pectral range of olar light from.4 to.7 m that photoynthetic organim are able to ue in the proce of photoynthei:.7m PAR.4m d [3.] EXAMPLE: Convert between radiance in per wavelength to radiance per wavenumber unit at = m. Given = 9.9 W m - r - m -. What i? = 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., an atmophere. General definition: Extinction i a proce that decreae the radiant intenity, while emiion increae it. NOTE: ame name : extinction = attenuation 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. 8

9 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 and amount of radiatively active 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. or a mall volume V of infiniteimal length d and area A containing radiatively active matter, change in intenity along the path d i proportional to the amount of matter in the path. + d or extinction: or emiion: d d e d [3.3], d e, d [3.4] where e i the volume extinction coefficient LENGTH - and i the ource function. The ource function ha emiion and cattering contribution or only cattering. Generally, the volume extinction coefficient i a function of poition. expreed mathematically a e. NOTE: Volume extinction coefficient i often referred to a the extinction coefficient. 9

10 Extinction coefficient = aborption coefficient + cattering coefficient a,, [3.5] 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 in the path e.g., number concentration, ma concentration, etc.. 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 -. or intance: cm g -, m kg -, etc. f i the denity ma concentration of a given type of particle or molecul then k, k, [3.6] a, ka, where the k, k, and k a, are the ma extinction, cattering, and aborption coefficient, repectively. NOTE: L ue k for both ma extinction and ma aborption coefficient! Uing the ma extinction coefficient, the Beer-Bouguer-Lambert extinction law Eq.[ ] can be expreed a d ke d, d ke, d

11 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: cro ection i in unit area LENGTH f N i the particle or molecule number concentration LENGTH -3 of a certain type of particle or molecul then, N, N [3.7] a, a, where,, and a, are the extinction, cattering, and aborbing cro ection, 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, then e = e and or thi ca the Extiction law can be expreed a exp exp [3.8]

12 Optical depth can be expreed in everal different way: d k d N d [3.9] f in a given volume there are everal type of optically active particle each with i, then the optical depth can be expreed a: i * i i d d N d [3.] i i where i and N i i the ma concentration denitie and particle concentration of the i-th type of pecie. i i i 4. Differential and integral form of the radiative tranfer equation. Let conider a mall volume V of infiniteimal length d and area A containing radiatively active matter. Uing the Extinction law, the change lo plu gain due to both the thermal emiion and cattering of intenity along the path d i d Dividing thi equation by e d, we find d d d d [3.] Eq. [3.] 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 d e d, e, d '

13 Uing the above expreion for d, we can re-write Eq. [3.] a d d d d or a [3.] Thee are other form of the differential equation of radiative tranfer. Re-arranging term in the above equation and multiplying 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 Thu d exp exp d exp exp d exp exp d and, uing d e d, we obtain a olution of the equation of radiative, tranfer often referred to a the integral form of the radiative tranfer equation: exp exp d [3.3] 3

14 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 ca the ource function conit of thermal emiion and cattering or 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. or many application, the atmophere can be approximated by a plane-parallel model to handle the vertical tratification of the atmopheric compoition and tructure. 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 peci 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 co x z z y where denote the angle between the upward normal and the direction of propagation of the light beam or zenith angle and i the azimuthal angle. Uing d = dz/co, the radiative tranfer equation can be written a d z co dz z z [3.4a] 4

15 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 = co, we have, d [3.4b] d Eq.[3.4] can be olved to give the upward or upwelling and downward or downwelling intenitie for a planet atmophere which i bounded on two ite. Upward intenity i for or / Downward intenity i for or / uing that co= co/= and co =- z = z top Top = z z = Bottom * * igure 3.4 Schematic repreentation of the plane-parallel atmophere. NOTE: or downward intenity, i replaced by. 5

16 6 The radiative tranfer equation [3.4b] can be written for upward and downward intenitie: d d [3.5a] d d [3.5b] A olution of Eq.[3.5a] give the upward intenity in the plane-parallel atmophere: d exp exp * * * [3.6a] and a olution of Eq.[3.5b] give the downward intenity in the plane-parallel atmophere: d exp exp [3.6b]

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

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