1. Terminology

2. Terminology Radiant energy (Q): energy carried by photons (in J).
Radiant flux (f): similar to power (rate of expending Q per time), e.g. dQ / dt (in J s-1 = W).
Radiant flux density: radiant flux incident upon a surface area, e.g. df / dA (in W m-2).
Irradiance (E): radiant flux density from external source
Exitance or emmitance (M): radiant flux density from internal source

3. Terminology
Irradiance is a measure of the amount of incoming radiant flux density onto a plane in Watts m-2.
Exitance is a measure of the amount of radiant flux density leaving a plane in Watts m-2 .

4. Terminology
Radiant intensity (I): radiant flux (not radiant flux density) per unit solid angle leaving a source in a given direction, e.g. df / dw (in W str-1).
Radiance (L): radiant intensity per unit area of projection (in W str-1 m-2), e.g. dI / dA cos q
Lambertian: when radiance does not change as function of direction (more on this later)

5. Terminology

6. Solid angle Terminology Ω = A / r2
The angle that, seen from the center of a sphere, includes a given area on the surface of that sphere. The value of the solid angle is numerically equal to the size of that area divided by the square of the radius of the sphere
The maximum solid angle is ~12.57, corresponding to the full area of the unit sphere, which is 4*Pi.
Standard unit of a solid angle is the Steradian (sr). (Mathematically, the solid angle is unitless, but for practical reasons, the steradian is assigned.)
Ω = A / r2

7. Terminology

8. Terminology

9. Terminology L = d2ϕ / (d dS cos )
Radiance (L ) is essentially “brightness” (radiant flux) at different wavelengths leaving an extended area (as opposed to a point source) in a particular direction.
power emitted (dϕ) per unit of solid angle (d) and per unit of the projected surface (dS cos) of an extended widespread source in a given direction,  ( = zenith angle, = azimuth angle)
L = d2ϕ / (d dS cos )
It is the most important remote sensing radiometric measurement.
Radiance is measured in Watts per meter squared per wavelength per steradian (W m-2 m–1 sr -1 )
Radiance is a function of wavelength (l) (EM signals are polychromatic).

10. d 
Projected surface dS cos 

11. Point Source d dϕ dA
Point source r
Consider flux dϕ emitted from point source into solid angle d, where df and d very small
Intensity I defined as flux per unit solid angle i.e. I = dϕ/d (Wsr-1)
Solid angle d = dA/r2 (steradians, sr)

12. Distributed Source  dϕ 
Plane source dS
dS cos 
What about when we have a plane source rather than a point?
Element of surface with area dS emits flux dϕ in direction at angle  to normal
Radiance L is intensity (I = dϕ/d) in a particular direction divided by the apparent area of source in that direction i.e. flux per solid angle per unit area (Wm-2sr-1)
Projected area of dS is direction  is dS cos , so…..
Radiance L = (dϕ/d) / dS cos  = dI/dS cos  (Wm-2sr-1)

14. Terminology

15. Terminology

16. Terminology

17. Terminology

20. Interaction with the Target
Interactions with incident (I) electromagnetic radiation:
Absorption (A)
Transmission (T)
Reflection (R)
A + T + R = I

21. t + a + r = 1.0 (Kirchoff’s Law)
Energy is Conserved
Transmittance (t): amount of radiation that passes through a material (0 to 1.0)
Absorptance (a): amount of radiation that is absorbed as it passes through (0 to 1.0)
Reflectance (r): amount of radiation that is reflected or scattered away (0 to 1.0)
t + a + r = 1.0 (Kirchoff’s Law)
Emission processes, usually thermal and due to atomic or molecular excitation, will change this energy balance by introducing a fourth term, Emissivity (e).

22. Hemispherical Quantities
Direct
Diffuse

28. Natural surfaces somewhere in between
Reflectance
When EMR hits target (surface)
Range of surface reflectance behaviour
perfect specular (mirror-like) - incidence angle = exitance angle
perfectly diffuse (Lambertian) - same reflectance in all directions independent of illumination angle)
Natural surfaces somewhere in between
From

29. Reflectance
Figure 2.1 Four examples of surface reflectance: (a) Lambertian reflectance (b) non-Lambertian (directional) reflectance (c) specular (mirror-like) reflectance (d) retro-reflection peak (hotspot).
(a)
(b)
(c)
(d)
Real surfaces usually display some degree of reflectance ANISOTROPY
Lambertian surface is isotropic by definition
Most surfaces have some level of anisotropy
From:

30. Cosine Law
Radiance linked to law describing spatial distributions of radiation emitted by a blackbody with uniform surface temperature T
Total emitted flux = T4
Surface of blackbody has same T no matter what the viewing angle
Intensity of radiation MUST be independent of viewing angle
Flux per unit solid angle divided by true area of surface must be proportional to cos , where  is the viewing angle with respect to the surface normal

31. αs αv
elevation angle ( α) = 90-θ, some time, elevation angle is also called altitude

32. Solar incidence angle is the angle between the Sun and a line perpendicular to a surface. So solar zenith angle is the solar incidence angle for a horizontal surface. For a slope surface, they are different.
Azimuth angle definition difference: starts from north or from south:

33. Cosine Law X
Radiometer dA Y X
Radiometer  Y
dA/cos 
Case 1: radiometer ‘sees’ dA and the flux is proportional to dA
Case 2: radiometer ‘sees’ dA/cos  > A… but T is the same as in Case 1…
Therefore, the flux emitted per unit area at angle   to cos  so that the product of emittance ( cos  ) times the area emitting ( 1/ cos ) is the same for all 
This is basis of Lambert’s Cosine Law

34. Cosine Law
Observed intensity (W/cm2·sr)) for a normal and off-normal observer; dA0 is the area of the observing aperture and dΩ is the solid angle subtended by the aperture from the viewpoint of the emitting area element.
Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge.
Radiant intensity observed from an ideal, diffusely reflecting surface (Lambertian surface) is directly proportional to the cosine of the view angle with respect to the surface normal

36. Directional reflectance: BRDF
Reflectance of most real surfaces is a function of not only λ, but viewing and illumination angles
Described by the Bi-Directional Reflectance Distribution Function (BRDF)
BRDF of area A defined as: RATIO OF incremental radiance, dLe, leaving surface through an infinitesimal solid angle in direction (v, v), TO incremental irradiance, dEi, from illumination direction ’(i, i) i.e.
 is viewing vector (v, v) e.g., view zenith and azimuth angles
’ is illumination vector (i, i) e.g., illumination zenith and azimuth angles
After: Jensen, J. (2000) Remote sensing of the environment: an Earth Resources Perspective.

37. Directional reflectance: BRDF
Note that BRDF defined over infinitesimally small solid angles , ’ and  interval, so it cannot be measured directly
In practice it is approximated over finite angles and 
surface area A
surface tangent vector i
2-v v i
incident solid angle 
incident diffuse radiation
direct irradiance (Ei) vector 
exitant solid angle 
viewer
Configuration of viewing and illumination vectors in the viewing hemisphere, with respect to an element of surface area, A.
From:

38. Directional reflectance: BRDF
Spectral behavior depends on view and illumination angles
Change these angles and the apparent reflectance changes
View and illumination angles are related to size, shape and distribution of objects on surface (structure of surface)
Typically CANNOT assume surfaces are Lambertian (isotropic)
Modelled barley reflectance, v from –50o to 0o (left to right, top to bottom).
From:

39. Measuring BRDF Need multi-angle observations. Can do three ways:
multiple cameras on same platform… quite complex technically.
Broad swath with large overlap so multiple observations build up multiple view angles… but surface can change over time while the measurements are being made.
platform with pointing capability… technically difficult

41. Bowl shape Directional reflectance: BRDF
increased scattering due to increased path length

42. Bowl shape Directional reflectance: BRDF
increased scattering due to increased path length

43. Directional reflectance: BRDF
Hot Spot
mainly due to a shadowing minimum
resulting reflectance is higher

44. “Hotspot”

46. Albedo
Albedo is ratio of the amount of EMR reflected by a surface to the amount of incident radiation on the surface.

48. Albedo
Total radiant energy (both direct and diffuse) reflected in all directions from the surface i.e. ratio of total outgoing to total incoming
Defines lower boundary condition of surface energy budget - determines how much incident solar radiation is absorbed
Albedo is BRDF integrated over whole viewing/illumination hemisphere
directional hemispherical reflectance (DHR) - reflectance integrated over whole viewing hemisphere resulting from directional illumination
bi-hemispherical reflectance (BHR) - integral of DHR with respect to hemispherical (diffuse) illumination
DHR =
BHR =

49. Albedo Actual albedo lies somewhere between DHR and BHR
Broadband albedo, , can be approximated as
where p() is proportion of solar irradiance at ; and () is spectral albedo
p() is function of direct and diffuse components of solar radiation and so is dependent on atmospheric state
Therefore albedo is NOT an intrinsic surface property (although BRDF is)