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If the global positioning system functions through radio waves, but radio waves are reflected in the ionosphere, then how do these electromagnetic waves reach the satellites?

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GPS receivers work by receiving signals from the satellites - nothing needs to be sent.

The signals are sent from the satellites at microwave frequencies of 1.22 and 1.57 GHz.

There is indeed a problem with the ionosphere - so-called "ionospheric delay", which must be accounted for, but signals at these microwave frequencies do not get reflected (much lower radio frequencies would be), they just travel at a frequency-dependent speed. By including signals with multiple frequencies, a correction for this effect can be made.

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Microwaves are used for communicating with distant geostationary satellites because they have shorter wavelengths and pass straight through the atmosphere.
While radio waves have longer wavelengths and are reflected by the ionosphere. They are suitable for communicating with satellites in low orbit.
Check this popular article for the detailed answer.

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