Radiometric properties of media. Mathieu Hébert

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1 1 Radiometric propertie of media Mathieu Hébert

2 Aborbing media 2 Seawater i blue due to wavelength-dependent aborption Almot all media are aborbing. The trength of aborption depend on the thickne of the medium. Watercolor Gla culpture by Corban Walker

Opacity, or ma aborption coefficient, mean the ratio

3 Cae of the atmophere 3 Spectral aborbance of the atmophere Aborbing (opaque) Nonaborbing (clear) NB: Opacity, or ma aborption coefficient, mean the ratio of linear aborption coefficient to the ma denity of the medium.

Cae of the atmophere 4 Spectral tranmittance of the atmophere in the viible pectrum Clear Aborbing Attenuation of light i due to cattering (Rayleigh cattering, Mie

4 Cae of the atmophere 4 Spectral tranmittance of the atmophere in the viible pectrum Clear Aborbing Attenuation of light i due to cattering (Rayleigh cattering, Mie cattering) and aborption by ga molecule, water droplet, a well a dut,pollution particle... e-coure of UVED (Univerité Virtuelle Environnement & Développement durable, in french)

5 Aborption, cattering : baic concept 5 Free mean path (in m) : Average ditance traveled by photon before a fraction exp(-1) i aborbed /cattered F 0 0 h F(h) Aborption free mean path : Scattering free mean path : Extinction free mean path : a e 1 ( ) = F e F 0 Aborption / cattering Coefficient (in m -1 ) : invere of the free mean path Aborption coefficient : Scattering coefficient : Extinction coefficient : a = 1/ a = 1/ ε= a+ = 1/ e

Beer law for aborping media 6 = 0 = af ( x) F( x + dx) = F x af x dx F( x + dx) F x d dx F F x dx = af x Fe The tranmittance of a lice of aborbing medium exponentially decreae a a function of the

6 Beer law for aborping media 6 = 0 = af ( x) F( x + dx) = F x af x dx F( x + dx) F x d dx F F x dx = af x Fe The tranmittance of a lice of aborbing medium exponentially decreae a a function of the thickne h of the medium. / 0 T = F h F = e ah The linear aborption coefficient a (m -1 ) generally depend upon wavelength. It i related to the imaginary part of the complex refractive index n + iκ(λ): 4πκ( λ) a ( λ ) = λ ax Beer law

Weakly cattering media 7 The tranmittance of a lice of weakly cattering medium, with linear cattering coefficient (m -1 ) alo exponentially decreae a a function of the thickne h of the medium:

7 Weakly cattering media 7 The tranmittance of a lice of weakly cattering medium, with linear cattering coefficient (m -1 ) alo exponentially decreae a a function of the thickne h of the medium: h T λ = e λ For both aborbing and weakly cattering media, linear aborption and cattering coefficient are ummed. They form the linear extinction coefficient (m -1 ) ελ = a λ+ λ a h h T λ = e λ λ

8 Non-uniform medium 0 h F 0 F(h) 8 The aborption and cattering coefficient may depend upon poition in the medium. In thi cae, the tranmittance of the medium i h 0 T e ε = NB: the extinction coefficient ε, thereby the tranmittance T, generally depend upon wavelength: x dx h 0 T e ε λ λ = ( x, ) dx

Aborption, cattering : baic concept 9 The optical thickne τ (no unit) decribe the ability for light to cro a lice of medium with given thickne h F 0 0 h F(h) log F h, λ τλ =

9 Aborption, cattering : baic concept 9 The optical thickne τ (no unit) decribe the ability for light to cro a lice of medium with given thickne h F 0 0 h F(h) log F h, λ τλ = F λ 0 τ 1 τ 1 mean almot tranparent mean opaque The tranmittance of the medium i therefore T λ = e τ λ The optical thickne of a hand i horter for red light than for blue light

10 Tranmittance, denity, attenuation 10 Definition Expreion Tranmittance over ditance h Spectral tranmittance per unit length a t λ = e λ h ( λ, ) = ( λ) T h t Spectral optical denity per unit length d ( λ ) = t( λ) log 10 T ( h) ( λ) λ, = 10 d h Spectral optical denity ( λ ) = ( λ) D d h T ( h), 10 D ( λ) λ = Spectral attenuation per unit length att ( λ ) = t ( λ) 10log 10 T ( h) ( λ) /10 λ, = 10 att h Spectral attenuation ( λ ) = ( λ) ATT att h T ( h), 10 ATT ( λ)/10 λ =

11 Categorie of cattering media 11

12 Angular pectral cattering coefficient 12 Incident collimated beam Fi ( λ, ) Unit volume V +d dω θ ϕ I ( λ,, θϕ, ) The irradiance at the front plane of the volume i E(λ). The flux cattered in a mall olid angle dω in direction (θ,ϕ) define an intenity I (λ,,θ,ϕ) Angular pectral cattering coefficient: λ,, θϕ, = ( λ θϕ) ( λ,) V I,,, E

13 Angular pectral cattering coefficient 13 Incident collimated beam Fi ( λ, ) Unit volume V +d dω θ ϕ I ( λ,, θϕ, ) The irradiance at the front plane of the volume i E(λ). The flux cattered in a mall olid angle dω in direction (θ,ϕ) define an intenity I (λ,,θ,ϕ) Angular pectral cattering ditribution function (in r -1 ): f λ,, θϕ, = ( λ θϕ) ( λ ) I,,, F, 2 π π/2 where F (λ,) i the total cattered flux: F λ, = I λ,, θϕ, in θdθdϕ ϕ= 0 θ= π/2 For iotropic cattering: ( θϕ, ),I( θϕ, ) = I = F/ 4π and f ( θϕ, ) = 1/ 4 io π

14 Angular pectral cattering coefficient 14 Incident collimated beam Fi ( λ, ) Unit volume V +d dω θ ϕ I ( λ,, θϕ, ) The irradiance at the front plane of the volume i E(λ). The flux cattered in a mall olid angle dω in direction (θ,ϕ) define an intenity I (λ,,θ,ϕ) Spectral phae function: P λ,, θϕ, = ( λ θϕ) ( λ ) I,,, I, io ( λ θϕ ) = π ( λ θϕ) P,,, f,,, 4 where I io (λ,) i the intenity which would be cattered by an iotropic diffuer. For iotropic cattering: ( θϕ, ),P( θϕ, ) = 1

15 Phae function 15 The aniotropy factor g i defined a 1 2 π π/2 g = P( θ) co( θ) in θθϕ d d 4π ϕ= 0 θ= π/2 or, when the phae function i often uniform in aimuth g 1 π/2 = co in 2 P θ θ θ d θ θ= π/2 The Henyey-Greentein phae function, frequently ued in cattering model, i baed on thi aniotropy factor θ g = 0.1 g = 0.3 g = 0.5

16 Total cro ection of a particle 16 Incident collimated beam The irradiance at the front plane i E(λ). The flux cattered in a mall olid angle dω in direction (θ,ϕ) define an intenity I (λ,,θ,ϕ) F ( λ ) F ( λ) F ( λ ) = F ( λ) F ( λ) i t i The total cro ection of the particle at λ (in m²) i: σλ = F E ( λ) ( λ) For a medium made of randomly ditributed particle, low denity, the cattering coefficient i : ( λ ) = Nσλ where N i the number of particle per unit volume (volume concentration). NB: one may alo define aborption and exctinction cro ection.

17 Angular cro ection of a particle 17 Incident collimated beam F λ i dω θ ϕ I ( λθϕ,, ) The irradiance at the front plane i E(λ). The flux cattered in a mall olid angle dω in direction (θ,ϕ) define an intenity I (λ,θ,ϕ) The angular cro ection of the particle at λ i: (,, ) σ λθϕ = I ( λθϕ,, ) E ( λ) For a medium made of randomly ditributed particle, low denity, the cattering coefficient i : ( λθϕ,, ) = Nσ( λθϕ,, )

Scattering model for the atmophere 18 Mie Scattering Solution to Maxwell equation

r) Rayleigh Scattering Approximation of Mie cattering for particle much maller than

cro-ection : ( 2r) 5 6 2 2π n 1 σλ = 4 2 3 λ n + 2 Light cattered perpendicularly to

18 Scattering model for the atmophere 18 Mie Scattering Solution to Maxwell equation decribing the cattering of an electromagnetic wave by a pherical particle (any radiu r) Rayleigh Scattering Approximation of Mie cattering for particle much maller than the wavelength of the radiation (applie to ga molecule in the ky): r << λ Total cro-ection : ( 2r) π n 1 σλ = λ n + 2 Light cattered perpendicularly to the incident wave i totally polaried: 2 R M M R R R = Rayleigh cattering M = Mie cattering Opalecent gla

19 Mie and Rayleigh phae function 19 Rayleigh (unpolaried light) Mie Iotropic cattering

20 Radiance of the cattered light 20 Incident collimated beam ( λ, ) E Thin dik perpendicular to axi (dv = A dik d ) ϕ dω θ ( λ,, θϕ, ) di (direction of obervation) The cattered intenity in the obervation direction (θ,ϕ) i: ( λ,, θϕ, ) = ( λ, ) ( λ,, θϕ, ) ( λ, ) di f E dv Since the volume of the thin dik i ( λ θϕ ) = ( λ θϕ) dl,,, di,,, / A dik 1 = ( λ, ) f ( λ,, θϕ, ) E ( λ, ) d 4π = λ P λ θϕ E λ d (, ) (,,, ) (, ) dv = A d, one can define the radiance: dik (P = phae function)

21 Radiance of the cattered light 21 Incident collimated beam Thin dik perpendicular to axi (dv = A dik d ) ϕ ( λ, ) E dω θ ( λ,,, θϕ, ) dl, app ob ob (direction of obervation) At a poition ob along the axi, due to extinction of the cattered beam, one obtain the apparent radiance element ( λ θϕ ) = ( λ θϕ) dl, app,, ob,, dl,,, e ob ε x dx

Radiance of the cattered light 22 Incident collimated beam ( λ, ) E dω ϕ θ L ( λ, ob, θϕ, ) ob (direction of obervation) Finally, the apparent radiance perceived at ob i: (,

22 Radiance of the cattered light 22 Incident collimated beam ( λ, ) E dω ϕ θ L ( λ, ob, θϕ, ) ob (direction of obervation) Finally, the apparent radiance perceived at ob i: (, θϕ, ) = ( λ,,, θϕ, ) L dl, app ob obervation ditance, app ob ob ε( x) dx = λ, P λ,, θϕ, e E λ, d 0 ob NB: thi model applie in ingle cattering mode (very low denity of particle)

Apparent contrat of object 23 Contrat of an object:

backgnd at d Example of decreaing contrat a the

23 Apparent contrat of object 23 Contrat of an object: C0 = L object L L backgnd backgnd Backgrnd obj Backgrnd obj C0 C 0 = 1 Apparent contrat at a given ditance d: C d = L object at d L L backgnd at d backgnd at d Example of decreaing contrat a the viewing ditance d increae : (object look lighter from long ditance)

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: