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The basic idea is that the radar sends out pulses of radiation and sees what comes back. The radiation is in the microwave frequency range, and it turns out that water droplets are excellent at sending microwave radiation back to the radar. Snow is okay, but not as good. By measuring the amount of time it takes the signal to come back to the radar, the radar can determine the distance to the rain and snow.
But you already knew that liquid water absorbed microwaves, didn't you?
There is also information in the intensity of the returned signal. The power of the returned signal is proportional to the number of water droplets and is also proportional to the square of the volume of water in each droplet. So heavier rain, as a general rule, produces a stronger signal, although it's difficult to quantify the rain rate because there's an infinite number of combinations of drops and drop sizes that can produce a given return signal.
Radars are also sensitive to the frequency of the returned signal. This can be useful, because if the raindrops are moving toward or away from the radar, the signal that is returned to the radar has a frequency slightly different than the initial signal. This "Doppler shift" is proportional to the velocity, and by measuring the shift, the radar can determine the speed at which the individual drops are moving toward or away from the radar. This is different from measuring how fast a whole area of rain is moving. For example, if the surface wind is from the southwest, all of the raindrops are going to be moving toward the northeast at low levels, no matter which way the clouds and storm systems are moving. To see how the areas of precipitation are moving, one must look at a movie loop or compare radar scans from two different times.
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