ASEN/ATOC 5235 Remote Sensing of the Atmosphere and Ocean
Homework 2: Electromagnetic Radiation
Due: Friday Feb 2, 2001

Answer any 4 questions (additional questions for extra credit).

1. What is the ratio of the spectral radiances of black bodies at 300 K and 6000 K at:
   (a) 1 GHz
   (b) 1000 Ghz
   (c) 1 mm
   (d) 0.1 mm?

   
   
   

2. Show that, for a black body, the wavelength at which B_n is maximum is about 1.76 times greater than the wavelength at which B_l is maximum at the same temperature.

   
   
   

3. Derive the Wien displacement law
   l_m = 2897/T
   from the simplified approximation of Planck's Law
   B₁ = C₁ l^-5 [exp(-C₂/lT)]
   where C₁ = 2phc² and C₂ = ch/k_B

   
   
   

4. Compute the blackbody radiance emission Bl at 3.7 mm, 10.8 mm, and 10.7 GHz at a temperature of 280 K. These wavelengths are used for satellite remote sensing of surface temperature because they are in atmospheric absorption "windows". Calculate the temperature error corresponding to a 0.5% radiance error at 280 K. Assume that there is no emission by the atmosphere at these wavelengths.

   
   
   

5. Calculate the Doppler shift (in Hertz) of radar and laser beams backscattered by particles with radial speeds of 0.01, 0.1, 1.0, and 10 m s^-1. Choose the wavelengths of 0.69 mm, 1.04 mm, and 10.6 mm for the laser and 3 cm, 5 cm, and 10 cm for the radar. Compare these frequency shifts to the frequency of the carrier beam.

   
   
   

6. An infrared scanning radiometer aboard a meteorological satellite measures the outgoing radiation emitted from the Earth's surface at a wavelength of 10 mm. Assuming a transparent atmosphere, what is the temperature of the surface if the observed intensity is 9.8 J m^-2 s^-1 mm^-1 sr^-1?

   
   
   

7. A passive scanning microwave radiometer (SSM/I) has a zenith viewing angle of 53^o and frequency of 10.35 GHz. At this viewing angle and frequency, flat water at 20^o C has an emissivity 0.574 for vertical polarization and 0.266 for horizontal polarization. The emitted radiation is partially linearly polarized, and Stokes parameters U and V are zero. Calculate the two Stokes parameters IV and IH in W m^-2 sr^-2 mm^-1 using the Rayleigh-Jeans limit. Also calculate the degree of polarization (Hint: the regular Planck function is for emission of both polarizations, while you need to consider one component (V or H) at a time. How does the Rayleigh-Jeans formula for the brightness temperature change when you put in terms of IV or IH instead of the intensity I as usual (Hint: we want the brightness temperature of a blackbody to equal the physical temperature)).