SOME DEFINITIONS IN RADIOMETRY
Sensors on board aircraft or satellites measure and usually quantify the energy received
whereas eyes or photographic plates are merely analog receivers. A measurement unit
system is therefore required and this shall be defined here.
Defining a direction in space
The direction of a line through any point on the Earth's surface is defined by 2 angles:
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the zenith angle θ, between the zenith (point on the celestial sphere located on the observer's
ascending vertical) and the direction observed,
the azimuth angle ϕ between the North (on the local meridian) and the projection of the line
on the Earth's surface.
The height (altitude or elevation) is sometimes used instead of θ:
h = (π / 2) - θ,
θ varies along the vertical plane from 0 to π/2 (0° to 90°),
ϕ varies along the horizontal plane from 0 to 2 π (0° to 360°).
Solid angle
A solid angle dΩ delimits a cone in space: d Ω = dS / r2 (in steradians, Sr) where dS is the
area cut by the cone over a sphere of radius r the center of which is at the apex of the
cone (see figure 4).
The solid angle corresponding to all the space around a point equals 4π Sr. The solid
angle of a revolving cone for which the plane half-angle at the apex is a equals: Ω = 2π (1 cos α) Sr.
For an observer on Earth, the half-space formed by the celestial arch (in other words an
hemisphere) therefore corresponds to 2π Sr ( α = 90°).
Radiance*, Emittance, Irradiance
* We talk about the radiance of a source and the irradiance at an object (by a source). Be
careful: the Earth's surface which receives the irradiance of the Sun, acts like a source for the
sensor since it reflects a back part of the solar energy it receives.
The objects studied can either emit radiation (radiance, emittance) or be "illuminated" by a
source (irradiance). We will therefore require a series of definitions for each of these
terms. Before giving the definitions, here's a reminder of the notion of Power:
Power (measured in Watts): power is the quantity of energy emitted by an object per unit
of time in all directions or received by an object per unit of time from all directions.
Definitions on sources
Objects emitting electromagnetic waves.
a) Point source
Intensity: intensity is the power emitted by a point source A per solid angle
unit.
IA = dW/dΩ (in W.Sr-1)
If the intensity is the same in all directions, the source is called isotropic.
Whenever a source does not have the same power in all directions it is said
to be anisotropic.
This notion is rarely used in remote sensing, as the Earth's surface observed
by satellite is not a point source.
b) Extended source
Radiance: radiance (L) is the power emitted (dW) per unit of the solid angle
(dΩ) and per unit of the projected surface (ds cosθ) of an extended
widespread source in a given direction (θ).
L = d2W / (dΩ ds cosθ) (in W.Sr-1. m-2)
If radiance is not dependent on θ and ϕ, i.e. if is the same in all directions,
the source is said to be Lambertian. Ordinary, surfaces are rarely found to be
Lambertian.
This notion is very important as the energy measured by the sensor is
proportional to the radiance of the observed source (Earth's surface).
Emittance: emittance (M) is the power emitted (dW) per surface unit of an
extended widespread source, throughout an hemisphere. The radiance is
therefore integrated along all the directions of a half-space (over an
hemisphere).
M = dW / dS (in W.m-2)
The following relationship is applicable for a Lambertian surface:
M=πL
According to the definition of radiance:
d2W / dS = L cosθ d Ω
i.e.: dW / dS = M ∨= L cosθ dΩ
Now the element of the solid angle dΩ under which the surface element of a
sphere delimited by directions (θ, ϕ) (θ + dθ, ϕ) (θ + dθ, ϕ + dθ) and ( θ, ϕ +
dϕ) is:
dΩ = sinϕ dθ dϕ
Hence the integration over an hemisphere is expressed as follows:
2π
π /2
∫ ∫ cosθ sin θ dθ
M = L dϕ
0
0
As the first integral equals 2π and the second 1/2, the result is:
M = πL
This mathematical formulation simply demonstrates that although the solid
angle under which the upper hemisphere is viewed is 2π, emittance of a
Lambertian surface can be found by multiplying radiance by π. This can be
intuitively understood: radiance is defined per unit of visible surface: let's take
an element of a Lambertian constant, defined surface: the measurement of
the energy emitted by this object will decrease by cos θ like the projected
surface when the direction of observation departs from the surface normal.
Emittance is a major notion in remote sensing, as a surface element on the
Earth re-emits the energy received throughout the hemisphere above the
local horizontal plane.
Definitions on objects
Objects receiving electromagnetic waves (as opposed to sources)
Irradiance: this is the power received per surface unit from all directions of a
half space (hemisphere).
E = dW / dS (in W.m-2)
The element of the Earth's surface ds receives an irradiance E from the
upper half space and acts for the sensor as a source of radiance L along a
direction θ.
Remarks:
Why is radiance defined as "directional" and irradiance as "hemispheric" ?
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the sensor receives energy radiated by the source dS along a specific direction. Radiance is
therefore directional;
irradiance of the Earth's surface in the visible range is caused by the Sun. As the latter has a
precise position on the celestial arch, we could be led to think irradiance is directional here.
This is not at all the case for a simple reason: through the atmosphere which scatters sunlight
(more details will be given on scattering later), visible radiation reaches us not only from the
direction of the Sun but also from all directions in the upper hemisphere. This is why we can
see clearly along a shady street. Consequently, solar irradiance is the sum of all direct and
diffuse irradiance and is therefore hemispheric.
All the previous definitions can be given for a narrow wavelength range
centered around λ. They can be noted that:
L(λ), M(λ), E( λ).
Summary of Radiometric Terms
Radiant flux (W): the amount of radiant energy emitted,
transmitted, or received per unit time.
Radiant flux density (W/m2): radiant flux per unit area
Irradiance (W/m2): radiant flux density incident on a surface
Radiant spectral flux density (W m-2 mm-1): radiant flux density
per unit of wavelength interval.
Radiant intensity (W/sr): flux emanating from a surface per unit
solid angle.
Radiance (W m-2 sr-1): radiant flux density emanating from a
surface per unit solid angle
Spectral radiance (W m-2 sr-1 mm-1): radiance per unit wavelength
interval.
Radiant emittance (W/m2): radiant flux density emitted by a
surface.
Summary of radiometric terms
Radiant energy (J)
Add time
Radiant flux (J/S = W)
Hemispherical
Directional
Add area
Radiant flux density (W/m2)
Irradiance (incident)
Radiant emittance (emitted)
Add wavelength
Radiant spectral flux
density (w m-2 mm-1)
Add direction
Radiant intensity(W/sr)
Add area
Radiance (W m-2 sr-1)
Add wavelength
Spectral radiance
(W m-2 sr-1 mm-1)