#### Transcript Presentation for chapter 6

Presentation for chapters 5 and 6 LIST OF CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. Surfaces - Emission and Absorption Surfaces - Reflection Radiative Transfer in the Atmosphere-Ocean System Examples of Phase Functions Rayleigh Phase Function Mie-Debye Phase Function Henyey-Greenstein Phase Function Scaling Transformations Remarks on Scaling Approximations SURFACES - EMISSION AND ABSORPTION • Energy emitted by a surface into whole hemisphere - spectral flux emittance: Energy emitted relative to that of a blackbody • Energy absorped when radiation incident over whole hemisphere – spectral flux absorptance: • Kirchoff’s Law for Opaque Surface: v,2 , Ts v,2 , Ts 2. SURFACES - REFLECTION • Ratio between reflected intensity and incident energy – Bidirectional Reflectance Distribution Function (BRDF): • • Lambert surface – reflected intensity is completely uniform . Specular surface – reflected intensity in one direction • In general: BRDF has one specular and one diffuse component: SURFACE REFLECTION Analytic reflectance expressions Minnaert Formula , o n k 1 o k 1 This model obey principle of reciprocit y Lommel - Seeliger 2n , o o Transmission through a slab • Transmitance • Transimitted intensity leaving the medium in downward direction TRANSMISSION THROUGH A SLAB For collimated beam: • Transmitted intensity is s v s s I Fv cos cos o o e Fv cos oTd v, o , vt • Flux transmitted s v s Fvt Fv cos o e Td v, o , cos d TRANSMISSION THROUGH A SLAB • Flux transmitance is vt F Td v, o ,2 s Fv cos o s v e Td v, o , cos d RADIATIVE TRANSFER EQUATION dI v av ' I v 1 a B d p ' , I v ' d s 4 4 ' incident direction scattered direction RADIATIVE TRANSFER EQUATION • For Zero scattering dI v I v Bv T d s With general solution I P2 I P1 e P1 , P2 P2 dtBt e P1 t P , P2 RADIATIVE TRANSFER IN THE ATMOSPHERE-OCEAN SYSTEM mr 1 in the atmosphere and mr 1.34 in • The refractive index is the ocean. • • In aquatic media, radiative transfer similar to gaseous media In pure aquatic media Density fluctuations lead to Rayleigh-like scattering. • In principle: Snell’s law and Fresnel’s equations describe radiative coupling between the two media if ocean surface is calm. • Complications are due to multiple scattering and total internal reflection as below RADIATIVE TRANSFER IN THE ATMOSPHERE-OCEAN SYSTEM • Demarcation between the refractive and the total reflective region in the ocean is given by the critical angle, whose cosine is: • where • • Beams in region I cannot reach the atmosphere directly Must be scattered into region II first EXAMPLES OF PHASE FUNCTIONS • We can ignore polarization effects in many applications eg: • Heating/cooling of medium,Photodissociation of molecules’Biological dose rates • Because: Error is very small compared to uncertainties determining optical properties of medium. • Since we are interested in energy transfer -> concentrate on the phase function RAYLEIGH PHASE FUNCTION • Incident wave induces a motion (of bound electrons) which is in phase with the wave ,nucleus provides a ’restoring force’ for electronic motion • All parts of molecule subjected to same value of E-field and the oscillating charge radiates secondary waves • Molecule extracts energy from wave and re-radiates in all directions • For isotropic molecule, unpolarized incidenradiation: RAYLEIGH PHASE FUNCTION • Expanding in terms of incident and scattered angles: • Azimuthal-averaged phase function is: RAYLEIGH PHASE FUNCTION • By expressing in terms of Legendre Polynomials: • Asymmetry factor for Rayleigh phase function is zero (because of orthogonality of Legendre Polynomials): • Only non-zero moment is MIE-DEBYE PHASE FUNCTION • Scattering by spherical particles • Scattering by larger particles: -> Strong forward scattering – diffraction peak in forward direction! • Why? • For a scattering object small compared to wavelength: -> Emission add together coherently because all oscillating dipoles are subject to the same field MIE-DEBYE PHASE FUNCTION • For a scattering object large compared to wavelength: • All parts of dipole no longer in phase • We find that: • Scattered wavelets in forward direction: always in phase • Scattered wavelets in other directions: mutual cancellations, partial interference HENYEY-GREENSTEIN PHASE FUNCTION • A one-parameter phase function first proposed in 1941: • No physical basis, but very popular because of the remarkable feature: • Legendre polynomial coeffients are simply: • Only first moment of phase function must be specified, thus HG expansion is simply: