1. Presentation for chapters 5 and 6

2. LIST OF CONTENTS 1.Surfaces - Emission and Absorption 2.Surfaces - Reflection 3.Radiative Transfer in the Atmosphere-Ocean System 4.Examples of Phase Functions 5.Rayleigh Phase Function 6.Mie-Debye Phase Function 7.Henyey-Greenstein Phase Function 8.Scaling Transformations 9.Remarks on Scaling Approximations

3. SURFACES - EMISSION AND ABSORPTION Energy emitted by a surface into whole hemisphere - spectral flux emittance: Energy absorped when radiation incident over whole hemisphere – spectral flux absorptance: Kirchoff’s Law for Opaque Surface: Energy emitted relative to that of a blackbody

4. 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:

6. SURFACE REFLECTION Analytic reflectance expressions

7. Transmission through a slab Transmitance Transimitted intensity leaving the medium in downward direction

8. TRANSMISSION THROUGH A SLAB For collimated beam: Transmitted intensity is Flux transmitted

9. TRANSMISSION THROUGH A SLAB Flux transmitance is

10. RADIATIVE TRANSFER EQUATION

11. For Zero scattering

12. RADIATIVE TRANSFER IN THE ATMOSPHERE-OCEAN SYSTEM The refractive index is in the atmosphere and in 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

14. 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

15. 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

16. 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:

17. RAYLEIGH PHASE FUNCTION Expanding in terms of incident and scattered angles: Azimuthal-averaged phase function is:

18. 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

19. 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

20. 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

22. 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: