2 Some Definitions in Radiometry
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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: • 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. -1 IA = dW/dΩ (in W.Sr ) 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 = π L 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 M = L dϕ cosθ sinθ dθ ∫ ∫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" ? • 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 Add direction Radiant flux density (W/m2) Radiant intensity(W/sr) Irradiance (incident) Add area Radiant emittance (emitted) Radiance (W m-2 sr-1) Add wavelength Add wavelength Spectral radiance Radiant spectral flux (W m-2 sr-1 mm-1) density (w m-2 mm-1).