Radiometry – remote sensing via microwave emission

1. Radiometry – remote sensing via microwave emission Chris Allen (callen@eecs.ku.edu) Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

2. Outline Thermal radiation • Blackbody radiation and Planck’s law • Stefan-Boltzmann law • Emissivity, graybodies, selective radiators • Rayleigh-Jeans approximation Temperature • Brightness temperature • Apparent temperature • Antenna temperature Radiative transfer • Extinction (absorption and scattering) • Emission Apparent temperature of terrain

3. Thermal radiation Emission (or absorption) of radiated energy may be the result of an atom-bound electron transitioning from one energy level to another or from a change in a molecule’s vibrational or rotational mode. In atomic and molecular gases these energy transitions are represented by a set of energy levels forming discrete lines in the electromagnetic spectrum. In liquids and solids the interaction between particles increases the number of degrees of freedom complicating the radiated (and absorbed) spectral characteristics. The probability of interactions is related to the atom’s or molecule’s kinetic energy in its random motion. Since a substance’s kinetic (or heat) energy is represented by its absolute temperature, the radiated energy intensity increases with temperature.

4. Blackbody radiation and Planck’s law Using classical physics (mechanics), theories were put forth by Wilhelm Wien (1893) and Lord Rayleigh (1900) that, in a piecemeal fashion, agreed well with experimentally measured radiative emissions. While the Wein law was valid for shorter (optical) wavelengths, the Rayleigh law was valid for longer wavelengths. In 1905 Lord Rayleigh and Sir James Jeans offered a more complete theory was presented (Rayleigh-Jeans law) that again only agreed well with experimental measurements at long wavelengths. Max Planck’s blackbody radiation law (1901) accurately predicted the spectral intensity of electromagnetic radiation at all frequencies or wavelengths by incorporating quantum theory.

5. Blackbody radiation and Planck’s law

6. Notation for radiometric quantities

7. Planck’s blackbody radiation law A blackbody is an idealized, perfectly opaque material that absorbs all the incident radiation at all frequencies, reflecting none. To maintain thermal equilibrium, a blackbody is also a perfect emitter. where Bf = Blackbody spectral brightness, W m-2 sr-1 Hz-1 h = Planck’s constant = 6.63 × 10-34 J s f = frequency, Hz k = Boltzmann’s constant = 1.38 × 10-23 J K-1 T = absolute temperature, K c = speed of light, 3 × 108 m s-1 Note: Only two variables: frequency, f, and temperature, T.

8. Planck’s blackbody radiation law Can be rewritten in terms of wavelength, l, instead of frequency. More useful form for use with optical frequencies such as infrared and visible bands. where Bl = Blackbody spectral brightness, W m-2 sr-1 m-1 h = Planck’s constant = 6.63 × 10-34 J s l = wavelength, m k = Boltzmann’s constant = 1.38 × 10-23 J K-1 T = absolute temperature, K c = speed of light, 3 × 108 m s-1 Note: Only two variables: wavelength, l, and temperature, T.

9. Planck’s blackbody radiation law The frequency yielding the maximum spectral brightness, fm, is found by finding the frequency where dBf /df = 0. From this process we get At room temperature (T ~ 300 K) fm = 17.6 THz or l = 17 mm, in the infrared. At the sun’s surface (T ~ 5780 K) fm = 339 THz or l = 884 nm, in the infrared. Similarly, the wavelength yielding the maximum spectral brightness, lm, can be found by setting dBl/ dl = 0. At room temperature (T ~ 300 K) lm = 9.6 mm or 31 THz, in the infrared band. At the sun’s surface (T ~ 5780 K) lm = 498nm or 602 THz, green light in the visible band. Different results are obtained since the curve resulting from Planck’s law takes on a different shape in the frequency domain compared to the wavelength domain and the peak values differ.

10. Temperature dependence of emission

11. Solar spectral irradiance Sun Diameter: 1.39 x 106 km Distance from Earth: 150 x 106 km • ~ 0.5° (angle subtended) W ~ 67 sr (solid angle)

12. Stefan-Boltzmann law The total brightness B for a blackbody at a temperature T is which evaluates to where s is the Stefan-Boltzmann constant Note: the total blackbody brightness B increases as T4. For an object at room temperature (T ~ 300 K), B  146 W m-2 sr-1 For the sun (surface temperature ~ 5780 K), B  20 MW m-2 sr-1

13. Emissivity, graybodies, selective radiators Blackbodies transform heat into electromagnetic energy with perfect efficiency. Natural targets generally have lower efficiencies and are sometimes called graybodies. This reduced efficiency is termed emissivity, e, and is defined as the ratio of the observed brightness relative to that of a blackbody at the same temperature. Since B(q, f)  Bbb, then 0 e(q, f)  1 A selective radiator denotes a case where emissivity is frequency or wavelength dependent, e(f) or e(l).

14. Emissivity, graybodies, selective radiators

15. Rayleigh-Jeans approximation At low frequencies (i.e., hf << kT), the exponential term in Planck’s law can be approximated by its series expansion Such that A deviation of less than 1% from Planck’s law requires At room temperature (T  300 K) this approximation is valid for frequencies below 117 GHz.

16. Relating brightness to received power (1/5) Two antennas, At and Ar, separated by distance R and aligned for maximum directivity. The power received is, Pr = Sr Ar The power density, Sr [W m-2], may be expressed in terms of the radiation intensity, Ft [W sr-1], as Sr = Ft / R2 And Ft may have directional properties, i.e., Ft(q, f)

17. Relating brightness to received power (2/5) While the emitting source is an aperture of finite extent (At), it is treated as a point source (i.e., we are in the far-field). For an extended (non-point) source (like the sky or a surface), we can define brightness as the radiated power per unit solid angle per unit area B = Ft / At [W sr-1 m-2]

18. Relating brightness to received power (3/5) So the received power is P = Ft Ar / R2 P = B Ar At / R2 Or since Wt = At / R2 P = B ArWt Example: for the sun (B  20 MW m-2 sr-1) with Wt = 67 sr delivers about 1340 W m-2 to the Earth outside the atmosphere.

19. Relating brightness to received power (4/5) For an extended target with brightness B(q, f), the incremental received power from each differential solid angle dW is dP = Ar B(q, f) dW When the receive aperture is not aligned for maximum directivity dP = Ar B(q, f) Fn(q, f) dW where Fn(q, f) is the normalized radiation pattern of the receiving antenna.

20. Relating brightness to received power (5/5) To allow for spectrally-dependent brightness, we introduce spectral brightness, Bf(q, f). Thus the total power received by the aperture over a band-width Df, extending from frequency f to f+Df is where the ½ term reflects the fact that only half of the incident power is detected due to the polarization selectivity of the antenna.

21. Brightness-related quantities Spectral power, Pf, is the power received by the antenna in a 1-Hz bandwidth [W Hz-1] where

22. Brightness-related quantities Sf is the spectral flux density [W m-2 Hz-1] Consider the case where an antenna is observing a discrete source, such as a star, that subtends a solid angle WS much smaller than the antenna’s main-beam solid angle, WM, such that Fn(q, f) ~ 1 over WS. For this case Where Bfs is the spectral brightness of the source.

23. Relating power and temperature (1/3) From the Rayleigh-Jeans law we know that brightness and temperature are linearly related at RF and microwave frequencies. To apply this to radiometric measurements, consider the experiment illustrated below.

24. Relating power and temperature (2/3) For a blackbody enclosure at temperature T For narrowband operation, we assume Bf ~ constant over Df permitting From antenna theory we know that

25. Relating power and temperature (3/3) So which agrees exactly with the noise power from a resistor at temperature T Therefore the power-temperature relationship permits us to speak of temperatures rather than power or brightness. Example: for T = 300 K and Df = 1 MHz, P = k T Df = 4.1 fW (4.1 × 10-15 W) or -144 dBW (dB relative to 1 W) or -114 dBm (dB relatice to 1 mW).If R = 50 , then the output voltage will be Vrms = (R P) = 450 nV. If R = 1000 , then Vrms = 2 V.

26. Brightness temperature Having related received power, P, to temperature and recognizing that emissivity, e, reduces an object’s brightness, leads us to define an equivalent brightness temperature, TB where T is the absolute physical temperature.

27. Apparent temperature Energy from various temperature sources are incident on the antenna. The apparent radiometric temperature, TAP(q, f),is the blackbody-equivalent temperature distribution representing the brightness distributionBi(q, f) of the energy incident on the antenna.

28. Antenna temperature The antenna radiometric temperature, TA, is the resistor-equivalent temperature that would deliver the same output power. With TAP and TA related as

29. Antenna temperature The antenna temperature is the apparent temperature distribution, TAP(q, f), integrated over the 4p solid angle weighted by the antenna’s radiation function, Fn(q, f) For the case where an antenna is observing a discrete source with apparent temperature TS, such as a star, that subtends a solid angle WS much smaller than the antenna’s main-beam solid angle, WM, the antenna temperature is This expression ignores atmospheric losses and other sources.

30. Antenna temperature Illustration of the various contributions to an antenna temperature measurement of surface temperature, TB. For each observation angle (q, f) contributions from the following are found: Atmospheric self-emission, TUP and TDN Terrain emission, TB Scattered downward-emitted atmospheric emission, TSC The combination, TB + TSC, is attenuated by the atmospheric loss factor, La Note: extraterrestial radiation sources are ignored

31. Radiative transfer Radiometric remote sensing requires an understanding of radiative transfer to analyze the various contributions to the apparent temperature TAP. As radiation traverses a medium (e.g., atmosphere) two radiation-matter interaction processes are involved: extinction and emission As radiation impinges on a boundary (e.g., surface) two radiation-matter interaction processes are involved: surface scattering and surface emission Expressions representing each of these processes will be developed.

32. Extinction For a small cylindrical volume of cross-section dA, the loss in brightness dB as energy normally incident on the cylinder’s lower face traverses the thickness dr is where B = brightness, W m-2 sr-1 e = extinction coefficient in of the medium, Np m-1 and e = a + s a = absorption coefficient, Np m-1 s = scattering coefficient, Np m-1 Albedo, a a = s /e , 0 ≤ a ≤ 1 1-a = a /e

33. Extinction Passage through an incremental length Dr of material that absorbs and scatters produces a brightness reduction of where  is the optical thickness [Np] Applying the Rayleigh-Jeans law to convert brightness to temperature  TAP(r) and 

34. Emission For a small cylindrical volume of cross-section dA, the increase in brightness dB as energy normally incident on the cylinder’s lower face traverses the thickness dr is where Ja = absorption source function [W m-2 sr-1](recall that absorption = emission) Js = scattering source function [W m-2 sr-1](stray brightness scattered into r direction)Applying the albedo concept yields

35. Emission Applying the Rayleigh-Jeans law to convert brightness to temperature  TAP(r) yields  T(r)   where T(r) = physical temperature at r (r is a scalar) B(ri) = brightness incident from direction ri  (r; ri) = the phase function, that portion of energy scattered from direction ri into direction r(r and ri are vectors)

36. Apparent temperature of an absorbing and scattering medium Combining the effects of extinction and emission to find the change in apparent temperature after passage through a layer of thickness r yields For a scatter-free medium, a << 1 and TSC(r) 0 where For clear sky conditions, the Earth’s atmosphere is scatter free in the RF and microwave bands. For most weather conditions, scattering may be ignored below 10 GHz.

37. Apparent temperature of atmosphere and terrain To illustrate the application of these concepts, consider the case of a stratified atmosphere where the temperature and absorption coefficients vary only with height (vertical axis z). The polarized apparent temperature of terrain observed from a height H above the terrain at an incidence angle  is where La(; H) is the atmospheric loss factor representing the total loss through the atmospheric path from the ground to observation point H in the direction  relative to the surface normal TB(, ; p) is the brightness temperature of the terrain TSC(, ; p) portion of radiation impinging on the terrain from the upper hemisphere that is scattered by the surface in the direction  TUP(; H) is the net upward-emitted radiation from the entire atmospheric path from the ground to observation point H

38. Apparent temperature of atmosphere and terrain Upwelling atmospheric radiation From the scatter-free special case we get The TAP(0) term is neglected since it is not part of the TUP and

39. Apparent temperature of atmosphere and terrain Downwelling atmospheric radiation Similarly we find which includes emissions from all heights reduced by the absorption of all intervening layers to the terrain level Note:TUP and TDN may not be equal. However if a = 0 then TUP = TDN = 0.

40. Apparent temperature of atmosphere and terrain Upwelling radiation from terrain and atmosphere For a stratified atmosphere Focusing first on the loss term we define the atmospheric loss factorLa La 1 and for  = 0, La = 1 La may be expressed in decibels as

41. Apparent temperature of atmosphere and terrain Upwelling radiation from terrain and atmosphere TAP(; 0) is the apparent temperature of the terrain in direction  at the terrain-atmosphere boundary. It includes the terrain brightness, TB(),and the scattered downwelling radiation, TSC(). The downwelling radiation that is incident on the surface may include the atmospheric radiation as well as cosmic and galactic radiation originating beyond the atmosphere.

42. Apparent temperature of atmosphere and terrain Thus Note that TUP(; p) is azimuth independent (no  dependence), as was TDN(; p) while both terrain and scattered terms are azimuth dependent. Special casesFor a lossless case (a is very small) For a very lossy case (La is very large)

43. Smooth surface scattering and emission Specular reflection from a smooth, planar surface (Fresnel reflection) where Pi = incident power  = specular reflectivity Pr = reflected power 1 = incidence angle p = polarization state (p = h or v) For the specular surface it can be shown that the emissivity is related to the reflectivity as

44. Rough surface scattering and emission Scattering from a rough surface is characterized by the bistatic scattering cross-section per unitarea (q0, f0; qs, fs, p0, ps) [unitless] where (q0, f0) = direction of incident power (qs, fs) = direction of scattered power (p0, ps) = polarization state of incident and scattered fields The emissivity of a rough surface is

45. Rough surface scattering and emission Similarly the scattered temperature, TSC(q0, f0; p0) is where TDN(qs, fs) = atmospheric downward emission

46. Emissivity of a dielectric slab For a slab of thickness d suspended in air (e.g., aradome). The slab interacts with radiation through its effective reflectivity, e effective transmissivity, e effective absorptivity, ae Here “effective” refers to the steady-state solution that includes all multiple reflections within the slab. For incident power Pi the steady-state reflected and transmitted powers are

47. Emissivity of a dielectric slab Therefore the absorbed power is or where Since thermodynamic equilibrium is assumed, the effective emissivity, ee, must equal ae, hence Special cases: For an opaque material , i.e., e = 0, For a lossless material , i.e., ae = ee = 0, For a nonreflective material, i.e., e = 0,

48. Emissivity of a dielectric slab There are two approaches for determining the effective reflectivity, transmissivity, and emissivity for a slab: A coherent approach accounts for both the amplitudes and phases of the fields reflected in the medium while an incoherent approach considers the amplitudes only. Since the coherent approach depends on phase information, minor variations regarding the geometry and dielectric properties will significantly impact the results. Both approaches yield similar results, though the coherent approach may show oscillations with thickness or frequency.

49. Emissivity of a dielectric slab Coherent approach where 1 is the top-layer reflectivity L2 is the one-way loss through the slab 2’ is the modified phase constant in the slab

50. Emissivity of a dielectric slab Incoherent approach where 1 is the top-layer reflectivity L2 is the one-way loss through the slab