1. Remote Sensing in Geology, Siegal & Gillespie (class website)
Wednesday, 26 January 2010
Lecture 7: Lambert’s law & reflection Interaction of light and surfaces
Reading
2.4.3 – spectra & energy interactions (p ) in:
Remote Sensing in Geology, Siegal & Gillespie (class website)
Previous lecture: atmospheric effects, scattering

2. What was covered in the previous lecture
Friday’s lecture:
Atmospheric scattering and other effects
- where light comes from and how it gets there
- we will trace radiation from its source to camera
- the atmosphere and its effect on light
- the basic radiative transfer equation: DN = a·Ig·r + b
LECTURES
Jan Intro
Jan Images
Jan Photointerpretation
Jan Color theory
Jan Radiative transfer
Jan Atmospheric scattering previous
Jan Lambert’s Law today
Jan Volume interactions
Feb Spectroscopy
Feb Satellites & Review
Feb Midterm
Feb Image processing
Feb Spectral mixture analysis
Feb Classification
Feb Radar & Lidar
Feb Thermal infrared
Mar Mars spectroscopy (Matt Smith)
Mar Forest remote sensing (Van Kane)
Mar Thermal modeling (Iryna Danilina)
Mar Review
Mar Final Exam
Today
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
Specifically:
Reflection/refraction of light from surfaces (surface interactions)
the RAT law
Beer’s Law, Fresnel’s Law, Snell’s Law, Lambert’s Law
Refraction, refractive index
Reflection
Types of surfaces, including Lambertian
Scattering and scattering envelopes
Topographic effects and shade
Particle size effects 2

3. Light is reflected, absorbed , or transmitted (RAT Law)
The amount of specular (mirror) reflection is given by Fresnel’s Law
Fresnel’s law rs
(n-1) 2 + K2
(n+1) 2 + K2
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·sin1 =n2·sin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 Snell’s Law
n = c/v

4. Refraction through a prism: absorptivity k is a function of l

5. Everyone discovered Snell’s Law (1621)
Ibn Sahl (Baghdad, 984)
On Burning Mirrors and Lenses
Ptolemy ( AD)
Thomas Harriot, 1602
Willebrod Snel van Royen (Snell), 1621
Renée Descartes, 1637
Christian Huygens, 1678

6. Fresnel’s Law describes the reflection rs of light from a surface
Augustin Fresnel
Fresnel lens
Fresnel’s Law describes the reflection rs of light from a surface
rs =
(n -1)2 +K 2
n is the refractive index
K is the extinction coefficient
(n+1)2 +K 2
This is the specular ray
K is not exactly 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 =
4pK
K is the imaginary part of the complex index of refraction:
m=n-jK l

7. Fresnel’s Law… is more complicated than shown.
The full formulation accounts for variation in
angles i and e

8. 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.

9. - diffuse or Lambertian
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

10. These styles of reflection from a surface con-trast with scattering within the atmosphere
diffuse
reflection
forward scattering
Types of scattering envelopes
Uniform scattering
Forward scattering
Back scattering

11. Forward scattering/reflection in snow
When light encounters a grain of snow it may scatter from sharp corners, reflect from the grain surface, or be transmitted through the grain. The effect is that light penetrates into a snow field and appears to be reflected diffusely from the surface. , but the actual mechanisms involve mainly transmission and refraction . The signature of this process is the observation that the “reflected” light may be colored by the bottoms of objects on the snow – example, skis.
snow
Light escapes from snow
because the absorption
coefficient k in e-kz is small
This helps increase the
“reflectivity” of snow
Snow grain
ski
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?

12. How does viewing and illumination geometry
affect radiance from Lambertian surfaces?
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
Unit area
Illumination I i
I cos i
i is the incident angle
; I is irradiance in W m-2

13. 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 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

14. 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

15. How does viewing and illumination geometry
affect radiance from Lambertian surfaces?
Reflection I i r p
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

16. Lambertian Surfaces
Specular ray I i i r p
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

17. Rough at the wavelength of light Plowed fields
Lambertian Surfaces
Rough at the wavelength of light
Plowed fields
L= I cos i r p
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

18. the brightness of a snow field doesn’t depend on e, the exit angle
DN=231
239
231
231
239
231
222

19. Reprise: reflection/refraction of light from
surfaces (surface interactions) e i i
Specular ray
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

20. { Effect of topography is to change incidence angle i’ r L= I cos i’ p
For topography elements >> l and >> IFOV i’ r p
L= I cos i’ i {
This is how shaded relief maps
are calculated (“hillshade”)
Shadow

21. Effect of topography is to change incidence angle
For topography elements >> l and >> IFOV i’ r p
L= I cos i’ i
Image
intensity
Shadow
For a nadir view

22. “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 i
Variable shaded surfaces Shadowed Surface i’ 29

23. Confusion of topographic shading and unresolved shadows33

24. - particle size effects - interaction mechanisms
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

25. 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

26. Obsidian Spectra Finest Coarsest (Rock) Reflectance Wavelength (nm)
mesh
Rock

27. 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