1 EE 543 Theory and Principles of Remote Sensing Derivation of the Transport Equation
O. Kilic EE 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
O. Kilic EE 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)
O. Kilic EE 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.
O. Kilic EE 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
Brightness O. Kilic EE 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 ,
O. Kilic EE 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.
O. Kilic EE Antenna Temperature (Overall Antenna Effects) We derived Averaged temperature over the solid angle of receive antenna F n ( ) AA SUMMARY T AP TATA
O. Kilic EE 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
O. Kilic EE Main Beam and Sidelobe Effects SUMMARY
O. Kilic EE Main Beam Efficiency Ratio of power contained within the main beam to total power. SUMMARY
O. Kilic EE Effective Main Beam Apparent Temperature, Antenna temperature if the antenna pattern consisted of only the main beam. SUMMARY
O. Kilic EE Antenna Stray Factor SUMMARY Ratio of power contained within the sidelobes to total power.
O. Kilic EE Effective Sidelobe Antenna Temperature Antenna temperature if the antenna pattern consisted of only the sidelobes. SUMMARY
O. Kilic EE Antenna Temperature and Beam Efficiency SUMMARY
O. Kilic EE Overall Antenna Efficiency and Antenna Temperature Combine beam efficiency and radiation efficiency: SUMMARY Desired value Measured value
O. Kilic EE 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
O. Kilic EE Summary The accuracy of radiometric measurements is highly dependent on the radiation efficiency, and main beam efficiency, of the antenna. SUMMARY
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE Scattering Cross Section Definitions: Power Relations Differential Scattering Cross-section Scattering Cross-section Absorption Cross-section Total Cross-section SUMMARY
O. Kilic EE Differential Scattering Cross Section o i R SiSi SUMMARY (m 2 /St)
O. Kilic EE Scattering Cross-section SUMMARY (m 2 )
O. Kilic EE Absorption Cross-section SUMMARY (m 2 )
O. Kilic EE Total Cross-section albedo SUMMARY (m 2 )
O. Kilic EE Phase Function SUMMARY
O. Kilic EE Phase Function (2) SUMMARY
O. Kilic EE 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 aa ss P out P in length r r + r
O. Kilic EE Incident and Output Power Change in brightness (1) (2)
O. Kilic EE Conservation of Power Extinction: off-scattering + absorption self emission + scattering (3)
O. Kilic EE Extinction in the Cylindrical Volume Due to Scattering Phenomenon
O. Kilic EE Scattering Coefficient Let denote the particle density in the volume. N psps Unit: #/m (4)
O. Kilic EE Loss Due to Scattering (5) Using (1) in (5) (6) where
O. Kilic EE Extinction in the Cylindrical Volume Due to Absorption Phenomenon
O. Kilic EE Absorption Coefficient Define “absorption coefficient: Unit: #/m (7) (8)
O. Kilic EE Total Power Loss Define “total coefficient” or “extinction coefficient” Total power loss is given by: (9) (10)
O. Kilic EE Loss in Power - Summary Particle density in v Incident Brightness Due to scattering and absorption.
O. Kilic EE Gain in the Cylindrical Volume Due to Scattering Phenomenon 0 aa ss 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.
O. Kilic EE Scattering of Incident Radiation Along s’ Towards s (11) (12)
O. Kilic EE Collective Increase in Power (2) (13)
O. Kilic EE Collective Increase in Power (14a) (14b)
O. Kilic EE Gain in the Cylindrical Volume Due to Self Emission Define emission source function as power emitted per (Volume Steradian Hertz) as (15)
O. Kilic EE Total Increase in Power (16) Self emission scattering
O. Kilic EE Power Conservation From (3) i.e. Using (1), (2), (10) and (16) (17)
O. Kilic EE 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
O. Kilic EE 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
O. Kilic EE Self Emission Function (18)
O. Kilic EE Scalar Transport Equation – Based on Extinction Coefficient Only (19)
O. Kilic EE Optical Distance 0 s s0s0 s1s1 ds Dimensionless # Loss factor per length (20)
O. Kilic EE Transport Equation as a Function of Optical Distance Divide the transport equation in (19) by and express as a function of ; i.e. (21)
O. Kilic EE Transport Equation as a Function of Optical Distance and Albedo
O. Kilic EE Transport Equation as a Function of Optical Distance JsJs JaJa
O. Kilic EE Solution to Transport Equation (1)
Solution to Transport Equation (2) O. Kilic EE
Solution to Transport Equation (3) O. Kilic EE
Transport Equation Scaled to Temperature (1) O. Kilic EE
Transport Equation Scaled to Temperature (2) O. Kilic EE
Transport Equation Scaled to Temperature (3) O. Kilic EE
Solution to Transport Equation (Temperature Form) O. Kilic EE
Solution to Transport Equation O. Kilic EE
Low Albedo Case, a<<1 O. Kilic EE
Solution for Low Albedo O. Kilic EE
Upwelling Radiation (Observe the Medium from Above) O. Kilic EE Low Albedo Case, a <<1
Upwelling Radiation O. Kilic EE
Upwelling Radiation O. Kilic EE
Solution: Upwelling Radiation O. Kilic EE
O. Kilic EE
O. Kilic EE 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.
Solution O. Kilic EE
Solution O. Kilic EE
Solution O. Kilic EE
Solution O. Kilic EE
Solution O. Kilic EE
Downwelling Radiation from a Layer (Observe the Medium from Below) O. Kilic EE
Downwelling Radiation O. Kilic EE
Downwelling Radiation O. Kilic EE
Downwelling Radiation O. Kilic EE
Summary O. Kilic EE
Atmospheric Radiation O. Kilic EE
O. Kilic EE 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
Solution O. Kilic EE
Solution O. Kilic EE
Solution O. Kilic EE
O. Kilic EE Summary: Upwelling and Downwelling Radiation Low Albedo Case
Upwelling Radiation (Low Albedo) O. Kilic EE SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution, a (z) = ao
O. Kilic EE Downwelling Radiation (Low Albedo) O. Kilic EE SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution, a (z) = ao
Special Cases: (Low Albedo) O. Kilic EE 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
Apparent Temperature Inside the Medium O. Kilic EE SPECIAL CASES: Constant Medium Temperature, T(z) = T o Uniform Particle Distribution, a (z) = ao
Application to Homogenous Half Space O. Kilic EE (eg. Soil, sea, etc.)
Recall Conservation of Power O. Kilic EE
Homogenous Terrain Contributions O. Kilic EE 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)
Terrain Contribution O. Kilic EE Upwelling radiation from a half space with uniform temperature Recall that for an infinite layer of uniform temperature:
Terrain cont’ed O. Kilic EE
Atmospheric Contribution O. Kilic EE Downwelling atmospheric temperature
Total Apparent Temperature for a Homogenous Half Space O. Kilic EE
Apparent Temperature at Altitude, H O. Kilic EE
O. Kilic EE References Microwave Remote Sensing, F. T. Ulaby, et.al. Addison-Wesley