Lecture 7: Lamberts law

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Tuesday, 27 January
Lecture 7 Lamberts law reflectionInteraction
of Light and Surfaces
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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The amount of specular (mirror) reflection is
given by Fresnels Law
Light is reflected, absorbed , or transmitted
(RAT Law)
rs
Fresnels 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
Beers law (L Lo e-kz)
Snells law n1sin?1 n2sin?2
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 Snells Law
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Fresnel lens
Augustin Fresnel
Fresnels 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
(n1)2 K 2
This is the specular ray
K is not the same as k, the absorption
coefficient in Beers 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 mn-jK
4pK
l
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Complex refractive index

• n n i k
• Consider an electrical wave propagating in the x
  direction
• ExE0,xexpi(kxx-?t)
• kx component of the wave vector in the x
  direction 2p/l w circular frequency 2pn
  vc/n n? v speed in light in medium c
  speed of light in vacuum k2p/lwn/c
• Substituting,
• Ex E0,xexpi(w(nik)/cx-?t)
• Ex E0,xexp(iwnx/c-wkx/c-i?t)
• Ex E0,xexp-wkx/cexp(i(kxx-?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.
  

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Surfaces may be - specular -
back-reflecting - forward-reflecting -
diffuse or Lambertian
Smooth surfaces (rmsltltl) generally are specular
or forward-reflecting examples water,
ice Rough surfaces (rmsgtgtl) 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
Types of scattering envelopes
Forward scattering
Back scattering
Uniform scattering
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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?
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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
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How does viewing and illumination geometry
affect radiance from Lambertian surfaces?
Unresolved surface element exactly fills the IFOV
at nadir, but doesnt 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 i cos
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
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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
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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
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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 ei if
surface were smooth like glass
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Lambertian Surfaces Rough at the wavelength of
light Plowed fields
The total light (hemispherical radiance)
reflected from a surface is L r I cos i W m -2
Lambertian surface - L is independent of e
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the brightness of a snow field doesnt depend on
e, the exit angle
DN231
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239
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Reprise reflection/refraction of light from
surfaces (surface interactions)
e
Specular ray
i
i
Incident ray
Reflected light
amount of reflected light r I 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
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Effect of topography is to change incidence angle
For topography elements gtgt l and gtgt IFOV
i
L I cos i
i
This is how shaded relief maps are calculated
(hillshade)
Shadow
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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.
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Confusion of topographic shading and unresolved
shadows
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Next well 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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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
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Obsidian Spectra
Finest
Reflectance
Coarsest (Rock)
Wavelength (nm)
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