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Tuesday, 26 January 2010. Lecture 7: Lambert’s law & reflection Interaction of light and surfaces. Reading. 2.4.3 – 2.6.4 spectra & energy interactions (p. 13-20) in: Remote Sensing in Geology , Siegal & Gillespie (class website). Previous lecture: atmospheric effects, scattering.
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Tuesday, 26 January 2010 Lecture 7: Lambert’s law & reflectionInteraction of light and surfaces Reading 2.4.3 – 2.6.4 spectra & energy interactions (p. 13-20) in: Remote Sensing in Geology, Siegal & Gillespie (class website) Previous lecture: atmospheric effects, scattering
The amount of specular (mirror) reflection is given by Fresnel’s Law (n-1) 2 + K2 (n+1) 2 + K2 Light is reflected, absorbed , ortransmitted (RAT Law) rs Fresnel’s law rs = N = refractive index K = extinction coefficient for the solid rs = fraction of light reflected from the 1st surface Mineral grain Absorption occurs here Transmitted component Beer’s law: (L = Lo e-kz) Snell’s law: n1·sin1 =n2·sin2 z = thickness of absorbing material k = absorption coefficient for the solid Lo = incoming directional radiance L = outgoing radiance Light passing from one medium to another is refracted according to Snell’s Law
Fresnel lens Augustin Fresnel Fresnel’s Law describes the reflection rs of light from a surface rs = ---------------- n is the refractive index K is the extinction coefficient (n -1)2 +K 2 (n+1)2 +K 2 This is the specular ray K is not the same as k, the absorption coefficient in Beer’s law (I = Io e-kz) (Beer – Lambert – Bouguer Law) K and k are related but not identical: k = --------- K is the imaginary part of the complex index of refraction: m=n-jK 4pK l
Complex refractive index n* = n + i k Consider an electrical wave propagating in the x direction: Ex=E0,x·exp[i·(kx·x·-ωt)] kx = component of the wave vector in the x direction = 2p/l w = circular frequency =2pn; v=c/n* = n·λ v = speed in light in medium c = speed of light in vacuum k=2p/l=w·n*/c Substituting, Ex = E0,x·exp[i·(w·(n+i·k)/c·x·-ω·t)] Ex = E0,x·exp[(i·w·n·x/c-w·k·x/c-i·ω·t)] Ex = E0,x·exp[-w·k·x/c]·exp[(i·(kx·x·-ω·t))] If we use a complex index of refraction, the propagation of electromagnetic waves in a material is whatever it would be for a simple real index of refraction times a damping factor (first term) that decreases the amplitude exponentially as a function of x. Notice the resemblance of the damping factor to the Beer-Lambert-Bouguer absorption law. The imaginary part k of the complex index of refraction thus describes the attenuation of electromagnetic waves in the material considered.
Surfaces may be - specular - back-reflecting - forward-reflecting - diffuse or Lambertian Smooth surfaces (rms<<l) generally are specular or forward-reflecting examples: water, ice Rough surfaces (rms>>l) generally are diffuse example: sand Complex surfaces with smooth facets at a variety of orientations are forward- or back-reflecting example: leaves Reflection envelopes
forward scattering diffuse reflection These styles of reflection from a surface con-trast with scattering within the atmosphere Types of scattering envelopes Back scattering Forward scattering Uniform scattering
snow ski Forward scattering in snow Light escapes from snow because the absorption coefficient k in e-kz is small This helps increase the “reflectivity” of snow You can easily test this: observe the apparent color of the snow next to a ski or snowboard with a brightly colored base: What do you see?
Unit area Unit area The total irradiance intercepted by an extended surface is the same, but flux density is reduced by 1/cos i --- the total flux per unit area of surface is smaller by cos i How does viewing and illumination geometry affect radiance from Lambertian surfaces? Illumination I i I cos i i is the incident angle ; I is irradiance in W m-2
How does viewing and illumination geometry affect radiance from Lambertian surfaces? Unresolved surface element exactly fills the IFOV at nadir, but doesn’t off nadir – part of the pixel “sees” the background instead Viewer at zenith Viewer at viewing angle e angular IFOV Same IFOV For a viewer off zenith, the same pixel is not filled by the 1 m2 surface element and the measured radiance is L = r p-1 I cos icos e therefore, point sources look darker as e increases 1 m2 Viewer at zenith sees r p-1 I cos i W sr-1 per pixel
How does viewing and illumination geometry affect radiance from Lambertian surfaces? Resolved surface element - pixels are filled regardless of e. Viewer at zenith Viewer at viewing angle e angular IFOV Same IFOV For a viewer off zenith, the same pixel now sees a foreshortened surface element with an area of 1/cos e m2 so that the measured radiance is L = r p-1 I cos i therefore, point sources do not change lightness as e increases 1 m2 Viewer at zenith still sees r p-1 I cos i W sr-1 per pixel
r p How does viewing and illumination geometry affect radiance from Lambertian surfaces? Reflection I i R= I cos i e I cos i i is the incident angle ; I is irradiance in W m-2 e is the emergent angle; R is the radiance in W m -2 sr-1
r p Lambertian Surfaces Specular ray I i i L= I cos i e I cos i i is the incidence angle; I is irradiance in W m-2 e is the emergence angle; L is the radiance in W m -2 sr-1 Specular ray would be at e=i if surface were smooth like glass
r p L= I cos i Lambertian Surfaces Rough at the wavelength of light Plowed fields Lambertian surface - L is independent of e The total light (hemispherical radiance) reflected from a surface is L = r I cos i W m -2
the brightness of a snow field doesn’t depend on e, the exit angle DN=231 239 231 231 239 231 222
Reprise: reflection/refraction of light from surfaces (surface interactions) e Specular ray i i Incident ray Reflected light ° amount of reflected light = rI cos i ° amount is independent of view angle e ° color of specularly reflected light is essentially unchanged ° color of the refracted ray is subject to selective absorption ° volume scattering permits some of the refracted ray to reach the camera Refracted ray
r p Effect of topography is to change incidence angle For topography elements >> l and >> IFOV i’ L= I cos i’ i { This is how shaded relief maps are calculated (“hillshade”) Shadow
r p Image intensity Effect of topography is to change incidence angle For topography elements >> l and >> IFOV i’ L= I cos i’ i Shadow For a nadir view
i’ i Variable shaded surfaces Shadowed Surface “Shadow,” “Shade” & “Shading” Shadow – blocking of direct illumination from the sun Shading - darkening of a surface due to illumination geometry. Does not include shadow. Shade – darkening of a surface due to shading & shadow combined 29
Next we’ll consider spectroscopy fundamentals - what happens to light as it is refracted into the surface and absorbed - particle size effects - interaction mechanisms Light enters a translucent solid - uniform refractive index Light enters a particulate layer - contrast in refractive index
Surface/volume ratio = lower Light from coarsely particulate surfaces will have a smaller fraction of specularly reflected light than light from finely particulate surfaces Surface/volume ratio = higher
Obsidian Spectra Finest Reflectance Coarsest (Rock) Wavelength (nm) mesh Rock 16-32 32-42 42-60 60-100 100-150 150-200
Next lecture: 1) reflection/refraction of light from surfaces (surface interactions) 2) volume interactions - resonance - electronic interactions - vibrational interactions 3) spectroscopy - continuum vs. resonance bands - spectral “mining” - continuum analysis 4) spectra of common Earth-surface materials