In :

from __future__ import print_function
from IPython.display import Image
import numpy as np



a) The attached color Doppler image shows a velocity pattern measured by Doppler radar with a wavelength of $\lambda=10$ cm and a PRF of 2200 Hz. Using the space below the figure, plot the wind direction (with wind from the east at 90 degrees and wind from the west at 270 degrees) and the wind speed as functions of height, Assume that the outer range ring is at vertical elevation of 24 km and that the radar is at sea level, and pretend that the wind speed is in m/s, not knots. (so the blue colors start at -51 m/s and go in steps of 7 m/s towards zero then on to speeds of +51 m/s in the yellow).

a) Answer -- range rings at elevations of 12 km and 24 km, so wind is blowing west to east with a maximum at 18 km of about 30 m/s

b) Now assume that the the PRF is reduced to 1100 Hz. Describe qualitatively what the radar image would look like if it was not corrected for aliasing. You can draw contour lines on the map and verbally describe the color patterns inside the ring.

b) Answer: first find mrmax and MUR for the two PRFs



In :

the_lambda=0.1
c=3.e8
PRF=2200
mrmax=the_lambda*PRF/4.
MUR=c/(2.*PRF)*1.e-3
print("At a PRF of {} Hz, the mrmax is {} m/s and the MUR is {} km".format(PRF,mrmax,MUR))




In :

PRF=1100
mrmax=the_lambda*PRF/4.
MUR=c/(2.*PRF)*1.e-3
print("At a PRF of {} Hz, the mrmax is {} m/s and the MUR is {} km".format(PRF,mrmax,MUR))



So there will be aliasing once the wind speed exceeds 27.5 m/s. The maxium wind speed of 30 m/s is to produce a phase angle of:



In :

delta_phi=30/27.5*180.
print("phase angle for 30 m/s is {} degrees".format(delta_phi))
print("radar will display this as {} degrees in opposite direction".format((360. - delta_phi)))
print("the magnitude of the aliased velocity will be {} m/s".format((360. - delta_phi)/180.*mrmax))



So the diagram will change color from blue to red or from red to blue at the 27.5 m/s isoline with the displayed windspeed decreasing from 27.5 to 25 m/s as it actually increases from 27.5 to 30 m/s

c) Find the smallest phase angle (magnitude and direction) between two pulses when the radial wind speed is 30 m/s and the PRF is at i) 1100 Hz and ii) 2200 Hz



In :

text="""At 2200 Hz, the mrmax is 55 m/s and the smallest phase angle for 30 ms is: {:5.2f} degrees
At 1110 Hz, the mrmax is 27.5 m/s and the smallest phase angle for 30 m/s is {:5.2f} degrees
"""
print(text.format((30/55.*180.),(360. - (30./27.5*180.))))
print(98.18*np.pi/180.)
print(163.64*np.pi/180.)



d) Find the maximum unambiguous range for this radar when the PRF is at i) 1100 Hz and ii) 2200 Hz.

As calculated above, at 2200 Hz the MUR is 68.1 km and at 1100 Hz the MUR is 136.6 km