Spring 2016 AGRON / E E / MTEOR 518
Problem Set 8: Due Tuesday, April 5, 2016.
Assigned March 22, 2016. Revised April 4 (Figure 2 and Figure 3).
See .pdf excerpt of UMF Chapter 5 on the syllabus.
1. Examine approximations for the temperature, pressure, and water vapor profiles in the atmosphere as given in UMF.
(a) If sea–level atmospheric temperature is To = 300 K, create a figure that illustrates the
temperature profile of the atmosphere up to 30 km using (5.1) in UMF. Plot temperature
(K) as a function of altitude (km).
(b) Assume that the pressure profile of the atmosphere is given by (5.6) in UMF. If sea–
level atmospheric pressure is Po = 1013 mbar, create a figure that illustrates the pressure
profile of the atmosphere up to 30 km. Plot pressure (mbar) as a function of altitude
(km). If you are curious, see how (5.5) in UMF compares to (5.6).
(c) Assume that the water vapor density profile of the atmosphere is given by (5.7) in UMF.
If sea–level atmospheric water vapor density is ρvo = 7.5 g m−3 , create a figure that
illustrates the water vapor density profile of the atmosphere up to 30 km for a scale
height of 2 km. Plot density (g m−3 ) as a function of altitude (km).
(d) What is the total mass of water vapor contained in a vertical column of this atmosphere
(the total precipitable water content of the atmosphere), in kg m−2 ? Note that 1 kg m−2
of precipitable water is equivalent to 1 mm of precipitable water in terms of the equivalent
depth of water per unit area (as we normally measure precipitation).
2. Now examine the absorption of microwave radiation by water vapor.
(a) Use (5.22) in UMF to approximate the total water vapor absorption coefficient of the
atmosphere. Then create a figure that plots the total water vapor absorption coefficient
as a function of frequency for 1 ≤ f ≤ 30 GHz at three levels in the atmosphere: sea–
level (z = 0 km); and two other higher levels of your choosing. Check your figure with
Figure 5.4 in UMF and Figure 1 (the subplot command puts multiple plots on the same
figure). I encourage you to use my MATLAB function kaH2O.m linked on the syllabus.
It will save you time and it is good programming form to modularize your code using
functions.
(b) According to Figure 1, does the total water vapor absorption coefficient increase or
decrease with altitude? Why?
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3. The optical depth, τ , quantifies the attenuation of microwave radiation.
(a) Plot τ (0, z) in Np as a function of altitude for 0 ≤ z ≤ 30 km at f = 10, 20, 22, and
30 GHz if water vapor is the only significant absorber.
(b) At what frequency is the optical depth of the atmosphere the largest?
(c) What is the “effective depth” of the atmosphere, i.e. the altitude at which τ (0, z)
essentially stops growing?
(d) Why does the optical depth not continue to get larger at higher altitudes?
4. Use a flat Earth approximation, assume that the atmosphere is composed of parallel layers
each with uniform properties, and truncate the atmosphere at 30 km (assume essentially all of
the atmosphere’s mass is below 30 km). Assume the atmosphere has a temperature, pressure,
and water vapor profile as defined by Problems 1a, 1b, and 1c.
(a) Create a figure that plots the brightness temperature of the atmosphere, TB , as a function
of frequency from 1 to 30 GHz for zenith angles of 0◦ , 30◦ , 60◦ , and 80◦ assuming that
water vapor is the only significant absorber. Curves for θ = 0 and 80◦ are shown in
Figure 2 to help you check your calculations.
(b) Qualitatively, what would the plot of TB versus frequency look like for a radiometer
observing a drier atmosphere (but same temperature profile)?
(c) Qualitatively, what would the plot of TB versus frequency look like for a radiometer
observing a warmer atmosphere (but same moisture content)?
(d) For a tropical atmosphere (warmer and wetter atmosphere), at what frequency will the
brightness temperature of the atmosphere be the largest?
5. Create a set of five theoretical atmospheres. Assume the same temperature and pressure
profile as in Problem 1a and 1b but vary the moisture content (total precipitable water
content) of each atmosphere by varying the sea–level water vapor density between 5 and
10 g m−3 . Two such atmospheres are shown in Figure 3, assuming that water vapor is the
only significant absorber in the atmosphere.
(a) Plot TB at a zenith angle of θ = 30◦ versus the total precipitable water content, with
separate lines illustrating the relationship between TB (θ = 30◦ ) and total precipitable
water content for f = 10, 22, and 30 GHz.
(b) How linear is the relationship between total precipitable water content and brightness
temperature?
(c) Which frequency would you use to measure the total precipitable water content of the
atmosphere?
(d) A typical radiometer has a precision of about 0.1 K. What is the smallest change in total
precipitable water that could be measured at the frequency that is most sensitive to the
water content of the atmosphere?
6. A new planet, P–518 (planet in sector 518), has been discovered. It’s atmosphere contains
only one gas, and it has been given the chemical symbol Ty in honor of the scientist who first
characterized the atmosphere (Dr. Tyler Antony). Ty gas has widely–spaced spectral lines,
with only a single line in the microwave region at 40 GHz. Absorption coefficients for Ty gas
at three atmospheric pressures are shown in the left half of Figure 4. A team of engineers
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(Amin Bandpy, Rehman Shahzad, and Pat Edmonds) has designed a microwave sounder to
monitor the temperature profile of the atmosphere of this new planet. The frequencies, and
corresponding weighting functions W (f, z), of this sounder are shown in the right half of
Figure 4. The performance of this sounder was tested in a simulation. In this simulation
P–518’s atmosphere was set to have a uniform temperature of 150 K. The results of this
simulation (brightness temperatures recorded by the sounder) are shown in Figure 5. After
further tests were conducted and passed, the sounder was fabricated and put into orbit about
P–518. The data recorded by the sounder are shown in Figure 6.
(a) Explain why Amin, Rehman, and Pat chose these frequencies for the sounder.
(b) Sketch the shape of the temperature profile of P–518’s atmosphere.
7. Read “Water dimer yields to spectroscopic study” with doi:10.1063/PT.3.1937.
(a) Water is responsible for what fraction of the atmosphere’s absorption of solar radiation
and longwave radiation emitted by Earth’s surface?
(b) What is it about water vapor absorption that hasn’t yet been understood?
(c) Describe the difference between a water dimer and monomer.
(d) How long did it take Tretyakov and colleagues to modify their equipment to make the
experiment work?
(e) What was the surprise?
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−1
10
0.045
water vapor attenuation coefficient, Np km−1
z = 0 km
z = 3.7 km
z = 7.4 km
z = 0 km
z = 3.7 km
z = 7.4 km
0.04
0.035
−2
10
0.03
0.025
−3
10
0.02
0.015
−4
10
0.01
0.005
5
10
15
20
frequency, GHz
25
0
30
0
10
20
frequency, GHz
Figure 1: Total water vapor absorption coefficient.
4
30
140
θ = 0°
θ = 80°
120
brightness temperature, K
100
80
60
40
20
0
0
5
10
15
20
25
30
frequency, GHz
Figure 2: Brightness temperature of the atmosphere as a function of frequency assuming water
vapor is the only significant absorber.
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precipitable water = 10 mm
precipitable water = 20 mm
40
brightness temperature, K
35
30
25
20
15
10
5
0
10
12
14
16
18
20
22
24
26
28
30
frequency, GHz
Figure 3: Brightness temperature of the atmosphere at θ = 30◦ assuming water vapor is the only
significant absorber for a “wet” atmosphere and a “dry” atmosphere.
5
10
1
60
p = 1013 mbar
p = 500 mbar
p = 200 mbar
f = 35.05 GHz
f = 38.05 GHz
f = 39.30 GHz
f = 39.75 GHz
f = 39.95 GHz
10
0
40
altitude, km
absorption coefficient, Np m -1
50
10
-1
30
20
10
10 -2
34
0
41
frequency, GHz
0
0.05
0.1
0.15
0.2
0.25
weighting function, km-1
Figure 4: At left, absorption coefficients for Ty gas. At right, microwave sounder frequencies
and weighting functions. Note that the x–axis tick marks in the left sub–figure correspond to the
frequencies of the microwave sounder.
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60
280
measured frequencies
260
50
240
brightness temperature, K
altitude, km
40
30
20
220
200
180
10
0
100
160
200
temperature, K
140
34
300
35
36
37
38
frequency, GHz
39
40
Figure 5: At left, temperature profile of P–518’s atmosphere for the simulation. At right, simulated
data collected by the microwave sounder.
60
280
measured frequencies
260
50
240
30
brightness temperature, K
altitude, km
40
?
20
220
200
180
10
0
100
160
200
temperature, K
300
140
34
35
36
37
38
frequency, GHz
39
40
Figure 6: At right, actual data collected by the microwave sounder orbiting P–518.
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