1. 1 EE 543 Theory and Principles of Remote Sensing Derivation of the Transport Equation

2. O. Kilic EE 543 2 Theory of Radiative Transfer We will be considering techniques to derive expressions for the apparent temperature, T AP of different scenes as shown below. Atmosphere Terrain TATA T UP TATA Terrain could be smooth, irregular, slab (such as layer of snow) over a surface. STEP 1: Derive equation of radiative transfer STEP 2: Apply to different scenes

3. O. Kilic EE 543 3 Radiation and Matter Interaction between radiation and matter is described by two processes: –Extinction –Emission Usually we have both phenomenon simultaneously. Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption) Emission: medium adds energy of its own (through scattering and self emission)

4. O. Kilic EE 543 4 Mediums of Interest The mediums of interest will typically consist of multiple types of single scatterers (rain, vegetation, atmosphere,etc.) First we will consider a single particle and examine its scattering and absorption characteristics. Then we will derive the transport equation for a collection of particles in a given volume.

5. O. Kilic EE 543 5 Apparent temperature distribution Apparent Temperature (Overall Scene Effects) antenna F n (  ) TATA Atmosphere T AP (  ) T DN T SC T UP TBTB T B : Terrain emission T DN : Atmospheric downward emission T UP : Atmospheric upward emission T SC : Scattered radiation Terrain SUMMARY

6. Brightness O. Kilic EE 543 6 In radiometry, both point and extended source of incoherent radiation (e.g. sky, terrain) are of interest. Brightness is defined as the radiated power per solid angle per unit area, as follows: The unit for brightness is Wsr -1 m -2 Power per solid angle (W/Sr) Function of , 

7. O. Kilic EE 543 7 Apparent Temperature T AP (  ) is the blackbody equivalent radiometric temperature of the scene. Incident brightness Consists of several terms SUMMARY Similar in form to Planck’s blackbody radiation.

8. O. Kilic EE 543 8 Antenna Temperature (Overall Antenna Effects) We derived  Averaged temperature over the solid angle of receive antenna F n (  ) AA SUMMARY T AP TATA

9. O. Kilic EE 543 9 Antenna Efficiency Radiation Efficiency Beam Efficiency: –Contributions due to sidelobes are undesired. –Ideally one would design a radiometer antenna with a narrow pencil beam and no sidelobes. SUMMARY

10. O. Kilic EE 543 10 Main Beam and Sidelobe Effects SUMMARY

11. O. Kilic EE 543 11 Main Beam Efficiency Ratio of power contained within the main beam to total power. SUMMARY

12. O. Kilic EE 543 12 Effective Main Beam Apparent Temperature, Antenna temperature if the antenna pattern consisted of only the main beam. SUMMARY

13. O. Kilic EE 543 13 Antenna Stray Factor SUMMARY Ratio of power contained within the sidelobes to total power.

14. O. Kilic EE 543 14 Effective Sidelobe Antenna Temperature Antenna temperature if the antenna pattern consisted of only the sidelobes. SUMMARY

15. O. Kilic EE 543 15 Antenna Temperature and Beam Efficiency SUMMARY

16. O. Kilic EE 543 16 Overall Antenna Efficiency and Antenna Temperature Combine beam efficiency and radiation efficiency: SUMMARY Desired value Measured value

17. O. Kilic EE 543 17 Linear relation Bias = B 1 +B 2 -( B 1 +B 2 ) Slope Depends on sidelobe levels, antenna efficiency and temperature Depends on antenna efficiency SUMMARY

18. O. Kilic EE 543 18 Summary The accuracy of radiometric measurements is highly dependent on the radiation efficiency, and main beam efficiency, of the antenna. SUMMARY

19. O. Kilic EE 543 19 Theory of Radiative Transfer We will be considering techniques to derive expressions for the apparent temperature, T AP of different scenes as shown below. Atmosphere Terrain TATA T UP TATA Terrain could be smooth, irregular, slab (such as layer of snow) over a surface. STEP 1: Derive equation of radiative transfer STEP 2: Apply to different scenes SUMMARY

20. O. Kilic EE 543 20 Radiation and Matter Interaction between radiation and matter is described by two processes: –Extinction –Emission Usually we have both phenomenon simultaneously. Extinction: radiation in a medium is reduced in intensity (due to scattering and absorption) Emission: medium adds energy of its own (through scattering and self emission) SUMMARY

21. O. Kilic EE 543 21 Terminology for Radiation/Scattering from a Particle Scattering Amplitude Differential Scattering Cross Section Scattering Cross Section Absorption Cross Section Total Cross Section Albedo Phase Function SUMMARY

22. O. Kilic EE 543 22 Scattering Amplitudes and Cross Sections Brightness directly relates to power, and satisfies the transport equation. We will examine the effects of presence of scattering particles on brightness. O s r B(r,s) is a function of position and direction Function of 5 parameters: r: x, y, z s:  SUMMARY

23. O. Kilic EE 543 23 Scattering Amplitude Consider an arbitrary scatterer: Imaginary, smallest sphere D i EiEi o EsEs R The scatterer redistributes the incident electric field in space: SUMMARY

24. O. Kilic EE 543 24 Scattering Amplitude(2) o:  s,  s i:  i,  i f(o,i) is a vector and it depends on four angles. SUMMARY Unit vectors along the incident and scattered directions

25. O. Kilic EE 543 25 Scattering Cross Section Definitions: Power Relations Differential Scattering Cross-section Scattering Cross-section Absorption Cross-section Total Cross-section SUMMARY

26. O. Kilic EE 543 26 Differential Scattering Cross Section o i R SiSi SUMMARY (m 2 /St)

27. O. Kilic EE 543 27 Scattering Cross-section SUMMARY (m 2 )

28. O. Kilic EE 543 28 Absorption Cross-section SUMMARY (m 2 )

29. O. Kilic EE 543 29 Total Cross-section albedo SUMMARY (m 2 )

30. O. Kilic EE 543 30 Phase Function SUMMARY

31. O. Kilic EE 543 31 Phase Function (2) SUMMARY

32. O. Kilic EE 543 32 Derivation of the Radiative Transfer (Transport) Equation  v =  a  s Consider a small cylindrical volume with identical scatterers inside. The volume of the cylinder is given by: Base area 0 aa ss  P out  P in  length r r +  r

33. O. Kilic EE 543 33 Incident and Output Power Change in brightness (1) (2)

34. O. Kilic EE 543 34 Conservation of Power Extinction: off-scattering + absorption self emission + scattering (3)

35. O. Kilic EE 543 35 Extinction in the Cylindrical Volume Due to Scattering Phenomenon

36. O. Kilic EE 543 36 Scattering Coefficient Let  denote the particle density in the volume. N psps Unit: #/m (4)

37. O. Kilic EE 543 37 Loss Due to Scattering (5) Using (1) in (5) (6) where

38. O. Kilic EE 543 38 Extinction in the Cylindrical Volume Due to Absorption Phenomenon

39. O. Kilic EE 543 39 Absorption Coefficient Define “absorption coefficient: Unit: #/m (7) (8)

40. O. Kilic EE 543 40 Total Power Loss Define “total coefficient” or “extinction coefficient” Total power loss is given by: (9) (10)

41. O. Kilic EE 543 41 Loss in Power - Summary Particle density in  v Incident Brightness Due to scattering and absorption.

42. O. Kilic EE 543 42 Gain in the Cylindrical Volume Due to Scattering Phenomenon 0 aa ss  P out  P in  An increase in power is experienced when the particles scatter energy along s direction when they are illuminated from other directions; i.e.

43. O. Kilic EE 543 43 Scattering of Incident Radiation Along s’ Towards s (11) (12)

44. O. Kilic EE 543 44 Collective Increase in Power (2) (13)

45. O. Kilic EE 543 45 Collective Increase in Power (14a) (14b)

46. O. Kilic EE 543 46 Gain in the Cylindrical Volume Due to Self Emission Define emission source function as power emitted per (Volume Steradian Hertz) as (15)

47. O. Kilic EE 543 47 Total Increase in Power (16) Self emission scattering

48. O. Kilic EE 543 48 Power Conservation From (3) i.e. Using (1), (2), (10) and (16) (17)

49. O. Kilic EE 543 49 Scalar Transport Equation Loss due to scattering and absorption Gain due to scattering of other incident energy along s direction Gain due to self emission

50. O. Kilic EE 543 50 Remarks on Emission Source Function  a ; absorption coefficient #/m Brightness of each particle inside the medium Power/(St. Area. Hz) Physical temperature Directly proportional to absorption

51. O. Kilic EE 543 51 Self Emission Function (18)

52. O. Kilic EE 543 52 Scalar Transport Equation – Based on Extinction Coefficient Only (19)

53. O. Kilic EE 543 53 Optical Distance 0 s s0s0 s1s1 ds Dimensionless # Loss factor per length (20)

54. O. Kilic EE 543 54 Transport Equation as a Function of Optical Distance Divide the transport equation in (19) by  and express as a function of  ; i.e. (21)

55. O. Kilic EE 543 55 Transport Equation as a Function of Optical Distance and Albedo

56. O. Kilic EE 543 56 Transport Equation as a Function of Optical Distance JsJs JaJa

57. O. Kilic EE 543 57 Solution to Transport Equation (1)

58. Solution to Transport Equation (2) O. Kilic EE 543 58

59. Solution to Transport Equation (3) O. Kilic EE 543 59

60. Transport Equation Scaled to Temperature (1) O. Kilic EE 543 60

61. Transport Equation Scaled to Temperature (2) O. Kilic EE 543 61

62. Transport Equation Scaled to Temperature (3) O. Kilic EE 543 62

63. Solution to Transport Equation (Temperature Form) O. Kilic EE 543 63

64. Solution to Transport Equation O. Kilic EE 543 64

65. Low Albedo Case, a<<1 O. Kilic EE 543 65

66. Solution for Low Albedo O. Kilic EE 543 66

67. Upwelling Radiation (Observe the Medium from Above) O. Kilic EE 543 67 Low Albedo Case, a <<1

68. Upwelling Radiation O. Kilic EE 543 68

69. Upwelling Radiation O. Kilic EE 543 69

70. Solution: Upwelling Radiation O. Kilic EE 543 70

71. O. Kilic EE 543 71

72. O. Kilic EE 543 72 Example Consider a downward looking, nadir pointing radiometer observing the ocean surface from an airborne platform above a 2 km thick cloud with water content of 1.5 g/m3. The absorption coefficient of the cloud is approximately given by: where f is in GHz and m v is the water content in g/m3. Assuming that the ocean has an apparent temperature, T AP (0,0) of 150 K, calculate the apparent temperature observed by the radiometer at f = 1 GHz. The cloud may be assumed to have a physical temperature of 275K.

73. Solution O. Kilic EE 543 73

74. Solution O. Kilic EE 543 74

75. Solution O. Kilic EE 543 75

76. Solution O. Kilic EE 543 76

77. Solution O. Kilic EE 543 77

78. Downwelling Radiation from a Layer (Observe the Medium from Below) O. Kilic EE 543 78

79. Downwelling Radiation O. Kilic EE 543 79

80. Downwelling Radiation O. Kilic EE 543 80

81. Downwelling Radiation O. Kilic EE 543 81

82. Summary O. Kilic EE 543 82

83. Atmospheric Radiation O. Kilic EE 543 83

84. O. Kilic EE 543 84 Example Atmospheric water vapor absorption coefficient at 22 GHz is where  v =m o e -0.5z is the water vapor density (g/m3), and T=T o -6.5z is temperature (K) and z is the altitute (km). Assuming that the most of the atmospheric absorption will be for the lowermost 10 km, calculate the downwelling and upwelling radiation temperatures for nadir direction. Let T o = 300K and  o = 7.5 g/m3

85. Solution O. Kilic EE 543 85

86. Solution O. Kilic EE 543 86

87. Solution O. Kilic EE 543 87

88. O. Kilic EE 543 88 Summary: Upwelling and Downwelling Radiation Low Albedo Case

89. Upwelling Radiation (Low Albedo) O. Kilic EE 543 89 SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution,  a (z) =  ao

90. O. Kilic EE 543 90 Downwelling Radiation (Low Albedo) O. Kilic EE 543 90 SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution,  a (z) =  ao

91. Special Cases: (Low Albedo) O. Kilic EE 543 91 SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution,  a (z) =  ao If layer H is several optical depths thick;  ao H>>1

92. Apparent Temperature Inside the Medium O. Kilic EE 543 92 SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution,  a (z) =  ao

93. Application to Homogenous Half Space O. Kilic EE 543 93 (eg. Soil, sea, etc.)

94. Recall Conservation of Power O. Kilic EE 543 94

95. Homogenous Terrain Contributions O. Kilic EE 543 95 There are two contributing factors to radiation observed from above: (1)Scattering of downward radiation from the atmosphere (2)Refraction of upward radiation from ground (terrain contribution)

96. Terrain Contribution O. Kilic EE 543 96 Upwelling radiation from a half space with uniform temperature Recall that for an infinite layer of uniform temperature:

97. Terrain cont’ed O. Kilic EE 543 97

98. Atmospheric Contribution O. Kilic EE 543 98 Downwelling atmospheric temperature

99. Total Apparent Temperature for a Homogenous Half Space O. Kilic EE 543 99

100. Apparent Temperature at Altitude, H O. Kilic EE 543 100

101. O. Kilic EE 543 101 References Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley