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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.1 W.D. Philpot, Cornell University 2. ELECTROMAGNETIC RADIATION Electromagnetic (EM) radiation and its transfer from sources to objects or from objects to sensors is fundamental to remote sensing, and it is important that certain characteristics of this phenomenon be understood. For most of the passive systems that we consider, the sun is the primary source of EM radiation, but we will also consider emitted (thermal) radiation and active systems that provide their on radiation. In any case, EM radiation is energy and that energy might be in wave or particulate (i.e., photon or quantum) form. All electromagnetic radiation has wave properties; at all levels, radiation shows interference and diffraction. But studies also indicate that the energy carried by electromagnetic waves may, under certain conditions, be regarded as discontinuous rather than the continuously graded energy that would be expected from a wave. 2.1 Maxwell's Equations The fundamental description of Electromagnetic radiation begins with Maxwell's equations which describe propagating plane waves. Maxwell's equations are written: In the presence of charge In vacuum (free space) 0 ρ ∇⋅ = ε E 0 ∇⋅ = E (2.1) 0 ∇⋅ = B 0 ∇⋅ = B (2.2) / ∇× = −∂ E B t / ∇× = −∂ E B t (2.3) 0 0 0 / t ∇× εµ∂ B J+ E 0 0 / t ∇× =ε µ ∂ B E (2.4) where: E = electric field B = magnetic-induction field ρ e = electric charge density ε 0 = electrical permittivity of free space = 8.85x10 -12 coulomb 2 /Newton-meter 2 µ 0 = magnetic permeability of free space = 12.57x10 -7 weber/ampere-meter t = time · = divergence (a spatial vector derivative operator) = curl (a spatial vector derivative operator) For our purposes, the key point to be gleaned from these equations is the symmetry between the electric and magnetic fields and the fact that they are always coupled. Electric fields are generated by time varying magnetic fields and magnetic fields are generated by time varying electric fields. By convention we usually only treat electric fields knowing that there will be an associated magnetic field. 2.2 Wave Properties The general concept of remote sensing using EM radiation involves a source or transmitter, the medium through which the wave propagates, and a receiver. As implied in Eq, (2.5), one property of a wave is its frequency, f, the number of vibrations, oscillations or cycles that the wave makes each second (one cycle/second = one Hertz, abbreviated Hz). A wave's frequency is determined by the source. The time of one vibration, the temporal period or period, T, is related to the frequency by f = 1/T. A propagating wave will have a characteristic velocity
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Page 1: 2. ELECTROMAGNETIC RADIATION - CEE Cornellceeserver.cee.cornell.edu/wdp2/cee6100/6100_monograph/...Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.1 W.D. Philpot, Cornell

Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.1 W.D. Philpot, Cornell University

2. ELECTROMAGNETIC RADIATION Electromagnetic (EM) radiation and its transfer from sources to objects or from objects to sensors is fundamental to remote sensing, and it is important that certain characteristics of this phenomenon be understood. For most of the passive systems that we consider, the sun is the primary source of EM radiation, but we will also consider emitted (thermal) radiation and active systems that provide their on radiation. In any case, EM radiation is energy and that energy might be in wave or particulate (i.e., photon or quantum) form. All electromagnetic radiation has wave properties; at all levels, radiation shows interference and diffraction. But studies also indicate that the energy carried by electromagnetic waves may, under certain conditions, be regarded as discontinuous rather than the continuously graded energy that would be expected from a wave. 2.1 Maxwell's Equations The fundamental description of Electromagnetic radiation begins with Maxwell's equations which describe propagating plane waves. Maxwell's equations are written:

In the presence of charge In vacuum (free space)

0

ρ∇ ⋅ =

εE 0∇ ⋅ =E (2.1)

0∇ ⋅ =B 0∇ ⋅ =B (2.2)

/∇ × = −∂ ∂E B t /∇ × = −∂ ∂E B t (2.3)

0 0 0 / t∇× = µ ε µ ∂ ∂B J + E 0 0 / t∇× = ε µ ∂ ∂B E (2.4)

where: E = electric field B = magnetic-induction field ρe = electric charge density ε0 = electrical permittivity of free space = 8.85x10-12 coulomb2/Newton-meter2

µ0 = magnetic permeability of free space = 12.57x10-7 weber/ampere-meter t = time ∇· = divergence (a spatial vector derivative operator) ∇ = curl (a spatial vector derivative operator) For our purposes, the key point to be gleaned from these equations is the symmetry between the electric and magnetic fields and the fact that they are always coupled. Electric fields are generated by time varying magnetic fields and magnetic fields are generated by time varying electric fields. By convention we usually only treat electric fields knowing that there will be an associated magnetic field. 2.2 Wave Properties The general concept of remote sensing using EM radiation involves a source or transmitter, the medium through which the wave propagates, and a receiver. As implied in Eq, (2.5), one property of a wave is its frequency, f, the number of vibrations, oscillations or cycles that the wave makes each second (one cycle/second = one Hertz, abbreviated Hz). A wave's frequency is determined by the source. The time of one vibration, the temporal period or period, T, is related to the frequency by f = 1/T. A propagating wave will have a characteristic velocity

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.2 W.D. Philpot, Cornell University

that is dependent on the medium. If a source is emitting a frequency, f, and if the velocity of wave propagation is v, at the end of one second there will be f waves spread over a distance v × (1s) (i.e., distance = velocity × time). One wave will occupy a distance λ where λ = v / f. (Note that λ = vT.) This distance, the spatial period or wavelength, is the shortest distance between consecutive similar points on the wave (e.g., between consecutive crests or troughs). As an example, c = 2.99792458 × 108 m/s is the speed of light in a vacuum, and this is essentially the speed of light in the atmosphere. (We will use the approximation, c = 3 × 108 m/s.) At a wavelength of 0.5 µm (blue-green light), the frequency of the radiation is:

8 -6 14 = v/ = (3 * 10 m/s) / (0.5 * 10 m) = 6 x 10 Hz = 600 THzf λ (2.5)

Similarly, for microwave radiation with a wavelength of 1 cm, the frequency of the radiation is:

8 -2 10 = v/ = (3 * 10 m/s) / (1 * 10 m) = 3 x 10 Hz = 30 GHzf λ (2.6)

Figure 2.1: Electromagnetic waves consisting of transverse, oscillating electric (E) and Magnetic (B) fields. The case shown is plane polarized radiation with the E and B fields both oscillating within a plane. 2.3 The speed of light and the index of refraction Equations (2.7) and (2.8) are solutions to Maxwell's equations (Eqs. 2.1-2.4) provided that the wave speed is given by:

( ) 1/ 2c −ο ο= µ ε (2.7)

where: c = velocity of light in a vacuum µo = permeability of free space εo = permittivity of free space

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.3 W.D. Philpot, Cornell University

Similarly, the speed of light in a particular medium (e.g., water, glass, calcite) may be expressed in terms of the magnetic permeability and the electric permittivity of the medium:

( ) 1/ 2v −= µ ε (2.8)

where: v = velocity of electromagnetic waves µ = permeability of the medium ε = permittivity of the medium The permeability of the medium, μ, is a measure of how well the medium stores or conducts a magnetic field. Analogously, the permittivity of the medium, ε, is a measure of how well the medium stores or conducts an electric field. For any medium, ε = εr εo and µ = µr µo; where εr = relative permittivity, also called the dielectric constant, and µr = the relative permeability.

The index of refraction of a medium (water, glass, air) is a measure of the speed of light in that medium relative to the speed of light in a vacuum:

( )1/ 2

1/ 2r r

o o

cn nv

µε= = = = ε µ µ ε

(2.9)

Expressing the speed of light in this way emphasizes the role of the electrical and magnetic properties of the medium controlling the way the light interacts with and propagates in the medium. In general, except for ferromagnetic materials, the magnetic properties are usually negligible, µr ≈ 1, and thus µ ≈ µ0.

The full effect of the material on light propagation may be expressed in the complex index of refraction: m = n + iκ (2.10)

where n is the standard index of refraction and κ represents the effects of absorption. (κ is sometimes called the electrodynamic absorption coefficient.) Both m and κ are strongly dependent on wavelength.

In an absorbing medium the dielectric constant is also represented as a complex number: r = ' + i "ε ε ε (2.11)

where ε' and ε" are the real and imaginary part of the dielectric constant. For common materials the index of refraction and the dielectric constant are related by: 2

rm = ε (2.12)

which leads to the relationships:

2 2' = m - ε κ (2.13)

" = 2 mε κ (2.14)

In analogy to the absorption coefficient, the electrical conductivity of the material is related to the imaginary part of the dielectric constant:

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.4 W.D. Philpot, Cornell University

0 r = 2 "/ σ π ε ε µf (2.15)

This quantity is sometimes called the optical conductivity. Except for the case of ferromagnetic materials, µr ≈ 1.

2.4 Polarization Figure 2.1 illustrates a plane polarized wave in which the electric field oscillates in the vertical plane and the magnetic field oscillates in the horizontal plane. As long as the orientation is constant, the wave is said to be plane polarized regardless of the specific orientation. Radiation is frequently split into vertical and horizontal polarization, but this is an arbitrary choice. A wave that is linearly polarized at any intermediate angle can be expressed as the combination of two waves, one horizontally polarized and one vertically polarized, that are exactly in phase (ϕx − ϕy = 0, π, or − π) and of equal amplitude (Ex = Ey). That is, we may express a single wave as a vector combination of two orthogonal waves:

x 0 xE = E sin(kz+ t - )ω ϕ (2.16)

y 0 yE = E sin(kz+ t - )ω ϕ (2.17)

If the amplitudes are equal but the phase angles are not equal (ϕx ≠ ϕy) then the electric field will rotate about the z-axis and the radiation is said to be circularly polarized. (Circular polarization can be right-handed or left-handed depending on whether ϕx > ϕy or vice versa.) If the amplitudes are also unequal, the electric field is seen as tracing out an ellipse in time and is therefore said to be elliptically polarized.

The state of polarization can be described efficiently using the Stokes Vector:

2 20 x yS E E= + Total amplitude (2.18)

2 21 x yS E E= − Horizontal linear polarization (2.19)

2 0x 0y y xS = 2 E E cos( )ϕ − ϕ Linear polarization @ 45° (2.20)

( )3 0x 0y y xS = 2 E E sin ϕ − ϕ Left-handed circular polarization (2.21)

where ⟨ . . .⟩ denotes a time average.

Examples of Stokes vectors are given below. In each case, the Stokes vector has been normalized so that S0 = 1.

S0 S1 S2 S3 1 0 0 0 Random polarization 1 1 0 0 x-polarized 1 -1 0 0 y-polarized 1 0 1 0 +45° linear 1 0 -1 0 - 45° linear 1 0 0 1 Right-hand circular 1 0 0 -1 Left-hand circular 1 0.6 0 0.8 Right-hand elliptical

If a wave is completely polarized the sum of the amplitudes of all the states of polarization must equal to total amplitude, then:

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.5 W.D. Philpot, Cornell University

2 2 2 20 1 2 3S = S + S + S (2.22)

For a partially polarized wave, the degree of polarization is then defined as the fraction of the total power that is contained in the polarized components:

2 2 21 2 3

20

S S SS

+ +

Polarization is a critical component of radar and passive microwave systems and the detectors for these systems are always polarized. In optical and thermal systems, polarization is frequently not part of the design, however polarization by optical components (beam splitters, diffraction gratings, etc. may introduce polarization effects that can, in turn, introduce artifacts into the data. 2.5 Energy in Electromagnetic Waves Energy may be defined as a capacity to do work. Work and energy are measured with the units (e.g., a joule, where 1 joule = 1 watt-second = 6.242 x l018 electron volts). Power is the rate of doing work; it is commonly measured in watts, where 1 watt = 1 joule/second.

Because electromagnetic radiation is energy, an electromagnetic wave delivers energy to any object which it strikes. The transport of energy per unit time (i.e., the power), either to or across a sample area, is described by the Poynting vector,

oS = E x B/µ (2.23)

2.6 Quantum Nature of Radiation At the microscopic level, the energy carried by electromagnetic waves is quantized; it is in the form of discrete or discontinuous packets (quanta or photons), rather than being the smooth, continuous flow predicted by wave theory. The energy of a single photon or quantum is given by:

Q h= f (2.24)

where h = Planck's constant (6.63 x l0-34 joule-seconds = 4.14 x l0-15 electron volt-seconds) and f = frequency.

The wave motion is called sinusoidal, simple harmonic or harmonic. This is the waveform that is most commonly used to represent electromagnetic radiation, if for no other reason than virtually any wave shape can be synthesized by a superposition of sinusoidal waves. A general expression for a sinusoidal wave traveling in the +z direction is:

x 0E = E sin(kz + t - )ω ϕ (2.25)

y 0B = (E /c) cos(kz + t - )ω ϕ (2.26)

where: Ex = amplitude or displacement of the electric field E0 = maximum amplitude or displacement of wave By = amplitude or displacement of the magnetic field k = wave number = 2π/λ

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.6 W.D. Philpot, Cornell University

ω = angular frequency = 2π f t = time ϕ = phase angle or phase constant c = speed of light in a vacuum ≈ 3 x 108 m/s The quantum nature of radiation can be demonstrated by the photoelectric effect (Figure 2.2). A photoelectric cell consists of a metallic surface (cathode) and a positively charged collecting electrode (anode), sealed within an evacuated glass bulb. When light (radiation) strikes the cathode, electrons may be freed from the metal (ionization) and given a certain kinetic energy. These electrons will be attracted to the anode, and an electric current can be measured.

Figure 2.2: Photoelectic cell with a metal cathode.

According to wave theory, the vibrations of the incoming electromagnetic waves cause the electrons of the metallic atoms to vibrate with increasing amplitude until they acquire sufficient energy to break loose from the metal. If this were the case, the energy would be directly related to light intensity (wave amplitude) and independent of light frequency. It is found, however, that

1.The electrons were emitted immediately - no time lag and therefore no accumulation of energy.

2. Increasing the intensity of the light increased the number of photoelectrons, but not their maximum kinetic energy.

3. Red light did not cause the ejection of electrons, no matter what the intensity.

4. A weak violet light caused the ejection of only a few electrons, but their maximum kinetic energies were greater than those for intense light of longer wavelengths.

5. The most intense beam of red light did not yield a single electron from most metals, while the faintest blue light instantly produced a few. Blue light is higher frequency, and thus higher energy, radiation than red light.

evacuated glass cell cathode

anode (+)

photon

(hf) e-

freed electron

battery

ammeter

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.7 W.D. Philpot, Cornell University

The energy relations controlling the photoelectric effect on described by:

o maxQ = Q + KE (2.27)

where Q is the energy of the incoming light photon that strikes the metal cathode; Qo is that part of the photon energy that is used to free the electron from the metallic surface; and KEmax is the remainder of the photon energy, which is given to the electron in the form of kinetic energy. KEmax is the maximum kinetic energy that the electron can have outside of the metallic surface; in nearly all cases, the actual energy will be reduced because of internal collisions. 2.7 Directional distribution of radiation The general description of radiation necessarily includes a description of its spatial distribution. We will need formalism for describing natural light sources (e.g., the sun and sky) as well as the interaction with real, rough surfaces that emit and reflect radiation in all directions. As a further complication, we must be able to describe the light distribution in three dimensions. However, before developing solid geometric notation, we review the concepts of plane geometry. 2.8 Angular measures Plane angles are two-dimensional angles, measured in the familiar units of degrees, minutes and seconds. They are also measured in radians. One radian is the angle subtended by an arc of a circle whose radius is equal to the arc length (Figure 2.3). This can be written:

(Subtended Angle in radians) x (Radius) = (Arc Length)

where the radius and arc length are measured in the same units of distance while the subtended angle is measured in radians. In accordance with this relationship, there are 2π radians in a circle whose radius is R and circumference is 2πR. Since a circle also has 360°/2π rad, then there are 57.2958°/rad. A commonly used plane angle unit in remote sensing is the milliradian, which is 10-3 radians or approximately 0.06°.

1 radian

r

r

r

Figure 2.3: Plane angle

– Plane angles are measured in degrees or radians

– There are 360° or 2π radians in a circle 1 radian = 360°/2π = 57.2958°/rad 1 mrad = (0.001 or 10-3 radians) ≈ 0.06° – One radian is the angle subtended by an arc of a circle

whose radius is equal to the arc length. – Circumference = 2πr

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Philpot & Philipson: Remote Sensing Fundamentals EM Radiation 2.8 W.D. Philpot, Cornell University, January 03

Plane angles are often used to describe the length (and by implication, the area) of the smallest ground area viewed by a detector. In this context the distance is typically the altitude of the sensor and the angle of interest is very small (≤ 2-3 mrad) allowing us to use the small angle approximation. For angles that are sufficiently small (<5°), then sin θ ≈ tan θ ≈ θ in radians. Thus,

s = 2 h tan θ/2 ≈ h θ (2.28)

s

θh

h = altitudes = distanceθ = angle subtended

by s at altitude h

Figure 2.4: Plane angle to describe the minimum field of view.

Table 1: Comparison of resolution limits for the human eye and several commonly used satellite systems.

length (s) IFOV (θ) altitude (h) system 10 cm 1 mrad 100 m 1 cm 0.1 mrad 100 m 1 m 1 mrad 1 km

800 m 1 mrad 800 km (MODIS) 80 m 0.1 mrad 800 km (MSS) 9.6 m 0.012 mrad 800 km (SPOT) 1.02 m 0.0015 mrad 681 km (IKONOS) 0.63 m 0.0014 mrad 450 km (Quickbird)

3 cm 0.3 mrad 100 m (eye)

Plane angles are inadequate for describing distributions in 3 dimensions. For that we need to expand the concept of an angle. The usual coordinate system used for describing radiation in three dimensions is polar coordinates in which the distance from the origin is specified as r, the angle from the vertical (zenith) is θ, and the angle describing the horizontal direction (azimuth) is specified as ϕ. If plane angles are extended to three-dimensional cones of solid angles, radians become steradians (Figure 2.5). A solid angle, denoted by ω or Ω is defined as an area on the surface of a sphere, divided by the square of the radius of the sphere. For example, the solid angle subtended by an object at the sensor's aperture would be the cross-sectional area of the object as seen by the sensor, divided by the squared distance between the object and sensor. Similarly,

arc length (in units of r) = subtended angle (in radians) × radius r (any units

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2.9

given that the surface area of a sphere is 4πR2, there are 4π steradians in a sphere (ω = 4πR2/R2 = 4π), and 2π steradians in a hemisphere.

Figure 2.5: Geometry for the definition of solid angles. θ is the zenith angle, ϕ is the azimuth

angle.

The area subtended by an incremental solid angle dΩ can be defined as the area on a sphere of radius, r, with dimensions of (r dϕ) in the azimuthal direction and of (r sinθ dθ) in the zenith direction. The ratio of the area on the sphere, divided by the square of the radius of the sphere then yields the differential element of solid angle in the direction (θ, ϕ):

dΩ = 2

2r sin d d

rθ θ ϕ = sin θ dθ dϕ (2.29)

For small angles the solid angle may be approximated by the area subtended by the solid angle on a sphere of radius r divided by the square of the radius.

ω ≈ A / r2 (2.30)

Figure 2.6: Small-angle approximation of the solid angle.

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2.10

2.9 Basic Radiative Terms There are many terms commonly (and not so commonly) used to describe radiation. For now we will only consider the three most important terms for our purposes. These are described below in some detail.

Power or flux, Φ [watts = Joule sec-1] The quantity that is actually measured is the radiant flux, radiant power, or, simply, power. Denoted by Φ (or P in older notation), power is normally measured in watts milliwatts [mW] = l0-3 watts; microwatts [µW] = l0-6 watts).

Radiance, L [watts m-2 ster-1] The power falling onto a surface of area, dA, from the solid angle dΩ, in the direction ξ = (θ,ϕ), is defined as the radiance, L, at that surface. If the surface is perpendicular to the incoming radiation, then the radiance may be defined simply as:

dLdA d

Φ=

Ω (2.31)

If the surface is oriented at an angle, θ, to the incoming radiation, then we must adjust the relationship accordingly:

d dLdA d cos dA d

Φ Φ= =

′ Ω θ Ω (2.32)

Figure 2.7: Projection of the area onto the direction of the radiation

Typical units for radiance are [w m-2 sr

-1] or [mw cm-2 sr -1]. Radiance describes power

radiating from a defined area through a defined solid angle. Alternatively, radiance may describe the power incident on a defined area through a defined solid angle.

Irradiance, E [w m -2] Irradiance describes the radiant power received at any surface (e.g., reflecting surface; sensor's aperture or detector). The focus for irradiance is on the surface and the total radiation at that surface. The incident direction is not specified because, for irradiance, it doesn't matter where the radiation comes from, just how much arrives. The surface itself however, is of a fixed size and has some orientation. For example, we might be concerned with how much radiation is incident on a 1 m2 area of the earth from the sky. In this case the radiation is only arriving from the upward hemisphere, and the effective area of the surface varies as the cosine of the incident angle (as in Equation (2.32)). In this case, the irradiance at that surface is just the integral of the radiance incident over the upward hemisphere:

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2.11

/ 2 2

ddownwelling

upward 0 0irradiancehemisphere

E L cos d L cos sin d dπ π

θ= ϕ=

= θ Ω = θ θ θ ϕ∫ ∫ ∫ (2.33)

Since the integral is only over the upward hemisphere we are dealing with downwelling irradiance, Ed. If we are interested in the radiation from the lower hemisphere, there is an equivalent radiance that can be defined for the upwelling radiation:

2

uupwelling

downward / 2 0irradiancehemisphere

E L cos d L cos sin d dπ π

θ=π ϕ=

= θ Ω = θ θ θ ϕ∫ ∫ ∫ (2.34)

When the integral refers to radiation emanating from a surface, it is sometimes referred to as exitance or radiant exitance, M. It is the same quantity with a different label to indicate the direction in which the radiation is traveling.

There is one extremely important special case. When the radiance is constant with angle (think of a completely overcast sky or a sunlit wall painted a flat white), then the integral is easy to solve:

2

/ 2 0

E L cos sin d dπ π

θ=π ϕ=

= θ θ θ ϕ∫ ∫ 2

/ 2 0

L cos sin d dπ π

θ=π ϕ=

= θ θ θ ϕ∫ ∫

/ 2

2 L cos sin dπ

θ=π

= π θ θ θ =∫ πL (2.35)

If we are concerned with incoming radiation, then this describes a Lambertian source. If we are looking at reflective surface, then it is a Lambertian reflector or a Lambertian surface.

2.10 Radiance Invariance One of the fundamental, crucial properties of radiance is that it is invariant over a path in a vacuum. This is simply a statement of conservation of energy, but can be less than obvious when one is computing radiance in a specific problem. Consider Figure 2.8 in which a radiance detector with area Ar and field of view defined by the solid angle, Ωr is directed toward a source, a distance r away. Area AS of the source completely fills the field of view of the detector.

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2.12

Figure 2.8 : Geometry used to illustrate radiance invariance law. The radiance viewed by the detector is the power originating from the source that reaches the detector, Φ0, through the solid angle defined by the detector field of view, Ωr, and incident on the detector area, Ar:

or

r rL

(2.36)

Alternatively, from the point of view of the source, the power emitted that reaches the detector, Φr, is emitted from surface As into the solid angle Ωs:

rs

s sL

(2.37)

where Φr = Φs. From the definition of the solid angle, we have the relationships:

2r s = A / r Ω (2.38)

2s r = A / rΩ (2.39)

s s r rA = A Ω Ω (2.40)

Using Equation 2.41, together with Equations (2.38) and (2.39), it is clear that Lr = Ls.

Most observations are complicated by the fact that we are interested in the radiance on a surface of area Ar, emanating from a surface As neither of which need be perpendicular to the line connecting the two. In this case, consider the source first. The power passing through surface As (which is perpendicular to the direction of interest) into solid angle Ωr must be the same as that leaving surface A's into the same solid angle. The radiance invariance law then allows us to assign the distantly measured field radiance Lr to the radiance from surface A's. (This is only exactly true if there is no loss along the path r, but we'll deal with that later.) From Equation 2.38,

s r s s r s s sL = / A / (A' cos )Φ Ω ≈ Φ θ Ω (2.41)

. Similarly, the radiance on surface Ar, as given by Equation 2.37, can be written:

r s r r s r r rL = / A / (A' cos )Φ Ω ≈ Φ θ Ω (2.42)

Since Ls = Lr and Φ0 = Φr, then:

s s s r r r (A' cos ) = (A' cos )θ Ω θ Ω (2.43)

One interesting conclusion to be derived from this analysis is that, if the radiance emanating from a surface is the same in all direction then, as long as the surface fills the field of view of the

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2.13

detecting device, Ωr, then the radiance observed will be independent of the angle of the emitting surface. It will also be independent of the distance, r, again, as long as the surface fills the field of view of the detector. The import of this fact is made clear by Mobley1:

"Consider the pleasant situation of sitting in front of a fireplace as you read this book. You look into the fire and perceive a certain "brightness" (in essence, radiance), and you feel a certain amount of heat coming from the fire (in essence irradiance; it is energy absorbed per unit time per unit area of your body that warms you). If you move your chair farther away from the fire, it appears smaller but the flames are just as bright; this is the radiance invariance law in action. However, you feel less heat coming from the fire; this is a consequence of the inverse square law for irradiance. This simple situation holds true as long as our radiometer − in this case, our eye − can resolve a finite solid angle for the fire. If we move so far away from the fire that our eye sees it as a true point source of light, then our eye responds to the irradiance received from the source. Thus the fire eventually fades from view as we continue to move farther away."

2.11 Black body radiation All real objects emit radiation with an intensity and spectral distribution that is determined by their temperature and their ability to emit efficiently. A blackbody is an ideal object that emits (and absorbs) radiation with perfect efficiency. This is true for all angles of incidence and for radiation of all wavelengths. A blackbody emits the maximum possible radiation that any object can emit in every direction, at every temperature and wavelength. The spectral distribution of blackbody radiation is given by Planck's radiation formula2:

( )

15

2

2 cMexp c / T 1λ

π=

λ λ − (2.44)

where: Mλ = spectral exitance or power emitted per unit area at wavelength λ c1 = hc2 k = Boltzman's constant c2 = h/k T = absolute temperature h = Planck's constant λ = wavelength c = speed of light

1 Mobley, C. (1994) 1994. Light and Water: Radiative Transfer in Natural Waters, Academic Press. 2 Max Planck initially proposed this formula in October of 1900 based on his intuitive observation that it would both

match experimental data and reconcile the difference between two formula, one by Wein which was applicable for the short wavelength limit, and one by Lord Rayleigh which was effective for the long wavelength range. Plank quickly provided a theoretical explanation for this formula and presented his results in December of 1900, noting that he had to assume that energy was discontinuous (e.g., quantized). He initially believed that this was merely a necessary mathematical trick rather than a fundamental requirement. Certainly no one realized that the implications would go far beyond the theory of radiation. (For more on Max Plank, see http://www.max-planck.mpg.de/frameset_e.html)

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0.0 2.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 0.0 2.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 4.0

10

Exitt

ance

[mw

/cm

2 /µ]

wavelength (µ)

0°C 20°C

30°C 40°C

50°C

1

Figure 2.9: Wavelength distribution for a blackbody according to Planck's radiation law As shown by the graphs of Planck's equation in Figure 2.9, each temperature produces a different curve; the discontinuities between curves follow from the quantum nature of radiation. By integrating Planck's equation over all wavelengths, one obtains the total radiation emitted at a given temperature (i.e., the area under the appropriate exitance vs. temperature curve) and the Stefan-Boltzmann law: 4

totM = Tσ (2.45) where: σ = Stefan-Boltzmann constant (5.67 x 10-8 watts/m2 K4), and T = absolute temperature. The total radiant energy emitted by a blackbody is thus a function only of its temperature.

At any temperature or wavelength, a real object (i.e., non-blackbody) will emit only a fraction of the radiation that a blackbody would emit at the same temperature or wavelength. This fraction is the object's emissivity, ε, or spectral emissivity, ελ. Emissivity is a measure of how well an object emits radiation as compared to a blackbody, over all wavelengths,

tot tot = M of object / M of blackbodyε (2.46)

Thus, the radiant emittance of the graybody (a body with a constant emissivity over the range of wavelengths of interest) is expressed as:

4totM = Tε σ (2.47)

Taking the derivative of Planck's formula (Equation 2.45) with respect to wavelength and setting it equal to zero yields another important formula, called Wien's Displacement Law, which specifies the wavelength at which the emission will reach its maximum value:

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maxAT

λ = (2.48)

where T is the temperature in K and A is a constant equal to about 2.898 x 10-3 K m. A typical earth temperature would be ~300 K, which makes λmax ≈ 10 µ the wavelength of maximum emission from the earth surface. From Planck's curves (Figure 2.9), it is apparent that a blackbody and most natural objects at temperatures normally encountered on the earth's surface will emit maximum radiation near 10 µm. 2.12 Solar Radiation

The sun is the source of radiation for the bulk of the passive sensing systems used in remote sensing. Radiation from the sun approximates a blackbody source at ~5800 K at its surface. The equivalent blackbody source would have a maximum near 500 nm. The reference surface for solar emission photosphere with temperatures ranging from ~6600 K at the bottom of the photosphere to 4400 K at the top3. The actual peak emission is closer to ~480 nm4. The solar spectrum above the earth's atmosphere differs from a blackbody primarily because of absorption in the solar atmosphere which gives rise to a fine structure of absorption bands (Fraunhofer lines). Scattering and absorption by the earth's atmosphere - especially by water - removes considerably more radiation. The resulting spectrum differs substantially from a blackbody source (Figure 2.10).

Figure 2.10: The solar spectrum at the earth: outside the atmosphere and at the earth surface, compared to a 5250°C (5520 K) blackbody spectrum. (http://en.wikipedia.org/wiki/File:Solar_Spectrum.png)

3 http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html 4 http://rredc.nrel.gov/solar/spectra/am1.5/

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The total irradiance at the earth available from the sun, averaging over all wavelengths, is referred to as the "solar constant" with an average value of 1360 w/m2. The actual value fluctuates by about 6.9% per year, from 1412 w/m2 in January to 1321 w/m2 in July, due to the change in the earth's distance from the sun. It also fluctuates slightly (~0.2%) on decadal time scales5. Example 2.1: Computation of solar radiance and irradiance Much of what has been presented in this chapter can be illustrated by considering the radiation from the sun as it illuminates the earth-moon system. To a good approximation the sun may be considered a gray body with an effective temperature of 5800K and an emissivity of εsun = 0.99. Thus, the sun's total radiant exitance is:

Mtot = ε σ T4 = 6.35 x 107 W m-2

which corresponds to a wavelength of maximum emission of:

3

maxA 2.898 x 10 K m 500 nmT 5800K

−λ = = =

The radiation that we see from the sun is emitted in the photosphere, the surface of which will define for us the diameter of the sun, Rsun≈ 6.96 x 108 m. The total power radiated by the sun is obtained by multiplying the total radiant exitance by the total surface area of the sun:

Φsun = (4 π Rsun2) Msun = 3.87 x 1026 W

At the earth, a distance of De≈ 1.5 x 1011 m, this power is spread out over the surface area of a sphere of area 4πDe

2, making the irradiance at the earth (the solar constant):

Esun = Φsun / 4πDe2 = 1.37 x 103 W m-2

(Measured values of Esun can be found at http://www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant.)

Treating the sun as a disk, the solid angle subtended by the sun as seen from the earth is

Ωsun= π (Rsun)2 / (De)2 = 6.71 x 10 –5 sr The corresponding radiance observed by a detector with a field of view filled by the sun is then,

Lsun = Esun / Ωsun = εσT4/π ≈ 2.02 x 107 W m-2 sr-1

2.13 Summary of Radiation quantities Notation in radiometry has a long history, complicated by the fact that developments were going on simultaneously in different fields meaning with the result that the notation has varied somewhat with time as well as discipline. The table below summarizes the terms that we will use in this class and includes some of the old symbols together with the newer symbols.

5 http://acrim.com/TSI%20Monitoring.htm

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Symbols Units new old a. Radiant flux or radiant power Φ P watts

The time rate of flow of radiant energy. We may imagine (Joules/sec) the flow to be in the form of swarms of tiny colored particles -- photons. Each photon contains a well-defined amount (a quantum, hf) of radiant energy.

b. Irradiance (also: exitance, emittance: notated as M) E, M H, M watts/m2

Power per unit area. The radiant flux incident on (irradiance) or emitted by (exitance, emittance) a surface divided by the area of that surface. (See Planck's equation.)

d. Radiance L N watts/m2/sr Radiant flux per unit solid angle per unit projected area. – If a source is percieved/measured as an extended

source, radiated power is described by radiance, L. – Measure of radiant intensity (I) of many small, but

known portions of an extended source.

2.14 The Electromagnetic Spectrum As shown in Figure 2.10, levels of electromagnetic radiation can be arranged on scales of wavelength, λ, frequency, ν, or energy content, Q (Q = hν and ν = v/λ; in free space, v = c). The electromagnetic spectrum extends from highly energetic cosmic radiation to very low energy oscillations along transmission power lines. Its divisions represent regions within which a common body of experimental technique exists (common sources or detectors). These regions often overlap, for it is possible to produce and sense a given frequency by two or more methods.

The parts of the spectrum, wavelength regions or spectral bands used for electromagnetic remote sensing generally range from the ultraviolet through the microwave. The most commonly used wavelength units in the ultraviolet, visible and infrared regions are micrometers (1 µm = l0-6 meters), although nanometers (1 nm = l0-9 meters) may be used by certain investigators, particularly those working in the visible and near-visible wavelengths. Millimeters (1 mm= l0-3 meters) and centimeters (1 cm = l0-2 meters) are used to describe wavelengths in the microwave region, but here, frequency designations tend to be more common than wavelength designations. Frequencies may be in kilohertz (1 kHz = l03 Hz), megahertz (1 mHz = l06 Hz) or gigahertz (1 gHz = l09 Hz).

Although specific sensors will not be reviewed in this monograph, it should be realized that the sensors listed in Table 1.1 are designed or confined to collect radiation in specific spectral bands (Figure 2.11). Photographic films, for example, are not sensitive to wavelengths longer than about 0.9 mm, while glass camera lenses will not transmit wavelengths shorter than about 0.36 µm. Thermal infrared sensing is normally conducted in the 3 to 5 µm or 3 to 14 µm bands.

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Figure 2.10: Names ranges of the electromagnetic spectrum in energy, wavelength and

frequency scales.

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Figure 2.11: Regions of the electromagnetic spectrum used for remote sensing and the principal

sensors.

1mm 10cm 1m

passive microwave

active systems

1cm

Ka- band

X- band

C- band

S- band

L- band

K- band

Ku- band

UHF

P-band