Lecture 7: Lambert’s law & reflection Interaction of Light and Surfaces
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Transcript of Lecture 7: Lambert’s law & reflection Interaction of Light and Surfaces
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Lecture 7: Lambert’s law & reflection
Interaction of Light and Surfaces
Tuesday, 27 January
2.4.3 – 2.6.4 spectra & energy interactions (p.13 – 20), Remote Sensing in Geology, B S Siegal & A R Gillespie, 1980 -- available on class website
Previous lecture: atmospheric effects, scattering
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Fresnel’s law
rs = (n-1) 2 + K2
(n+1) 2 + K2
N = refractive indexK = extinction coefficient for the solidrs = fraction of light reflected from the 1st surface
rs
The amount of specular (mirror) reflection is given by Fresnel’s Law
Light is reflected, absorbed , or transmitted (RAT Law)
Transmitted component
Absorption occurs here
Mineral grain
Light passing from one medium to another is refracted according to Snell’s Law
Snell’s law: n1·sin1 =n2·sin2
Beer’s law: (L = Lo e-kz)z = thickness of absorbing materialk = absorption coefficient for the solidLo = incoming directional radianceL = outgoing radiance
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Fresnel’s Law describes the reflection rs of light from a surface
rs = ----------------(n -1)2 +K 2
(n+1)2 +K 2
n is the refractive index K is the extinction coefficient
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 = ---------4K
K is the imaginary part of the complex index of refraction:m=n-jK
This is the specular ray
Augustin Fresnel Fresnel lens
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Complex refractive index
n* = n + i
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 = 2/; = circular frequency
=2v=c/n* = n·λ; v = speed in light in medium; c = speed of light in vacuum; k=2/=·*/c
Substituting,Ex = E0,x·exp[i·(·(n+i·)/c·x·-ω·t)]Ex = E0,x·exp[(i··n·x/c-··x/c-i·ω·t)]Ex = E0,x·exp[-··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 of the complex index of refraction thus describes the attenuation of electromagnetic waves in the material considered.
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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
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These styles of reflection from a surface contrast with scattering within the atmosphere
diffusereflection
forward scattering
Types of scattering envelopes
Uniform scattering Forward scattering Back scattering
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Forward scattering in snow
ski
Light escapes from snowbecause the absorption coefficient k in e-kz is small
This helps increase the“reflectivity” of snow
You can easily test this: observe the apparentcolor of the snow next to a ski or snowboard with a brightly colored base:What do you see?
snow
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How does viewing and illumination geometry affect radiance from Lambertian surfaces?
i
I
I cos i
Illumination
i is the incident angle; I is irradiance in W m-2
The total irradiance intercepted by anextended surface isthe same, but flux density is reduced by 1/cos i --- the total flux per unit area of surface is smaller by cos i
Unit area
Unit area
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How does viewing and illumination geometry affect radiance from Lambertian surfaces?
Viewerat zenith Viewer
at viewing angle e
Viewer at zenith seesr -1 I cos i W sr-1 per pixel
angularIFOV
Same IFOV
1 m2 For a viewer off zenith, the same pixel is not filled by the 1 m2 surface element and the measured radiance is
L = r L = r -1-1 I cos i I cos i cos ecos etherefore, point sources look darker as e increases
Unresolved surface element exactly fills the IFOV at nadir, but doesn’t off nadir – part of the pixel “sees” the background instead
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How does viewing and illumination geometry affect radiance from Lambertian surfaces?
Viewerat zenith Viewer
at viewing angle e
Viewer at zenith still seesr -1 I cos i W sr-1 per pixel
angularIFOV
Same IFOV
1 m2
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 L = r -1-1 I cos i I cos itherefore, point sources do not change lightness as e increases
Resolved surface element -pixels are filledregardless of e.
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How does viewing and illumination geometry affect radiance from Lambertian surfaces?
i
I
I cos i
Reflection
R= I cos ie
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
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i
I
I cos i
L= I cos ie
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
Lambertian Surfaces
Specular ray
i
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L= I cos ir
The total light (hemispherical radiance) reflectedfrom a surface is L = r I cos i W m -2
Lambertiansurface -L is independentof e
Lambertian SurfacesRough at the wavelength of light
Plowed fields
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DN=231
222
231
239
231
231239
the brightness of a snow field doesn’t depend on e, the exit angle
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ii
Reprise: reflection/refraction of light from surfaces (surface interactions)
Incident ray
Refracted ray
Specular rayReflected light
° amount of reflected light = rr I cos I cos ii° amount is independentindependent of view angle ee° 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
e
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Effect of topography is to change incidence angle
i
Shadow
i’
For topography elements >> l and >> IFOV
This is how shaded relief maps are calculated (“hillshade”)
L= I cos i’r
Imageintensity
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Shade vs. Shadow
Shadow – blocking of direct illumination from the sun
Shade – darkening of a surface due to illumination geometry. Does not include shadow.
i
Variable shaded surfaces Shadowed Surface
i’
29
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33Confusion of topographic shading and unresolved shadows
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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
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Light from coarselyparticulate surfaces will have a smaller fraction of specularly reflected light than light from finely particulate surfaces
Surface/volume ratio = lower
Surface/volume ratio = higher
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Obsidian Spectra
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
350 850 1350 1850 2350
Rock
16 - 32
32 - 42
42 - 60
60 - 100
100 - 150
150 - 200
Wavelength (nm)
Finest
Coarsest (Rock)
Ref
lect
ance
mesh Rock 16-32 32-42 42-60 60-100 100-150 150-200
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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