IEEE Spectrum's Can Astronauts Use GPS to Navigate on the Moon? NASA Scientists Say Yes says:
Kar-Ming Cheung and Charles Lee of NASA’s Jet Propulsion Laboratory in California did the math, and concluded that the answer is yes: Signals from existing global navigation satellites near the Earth could be used to guide astronauts in lunar orbit, 385,000 km away. The researchers presented their newest findings at the IEEE Aerospace Conference in Montana this month...
Cheung and Lee plotted the orbits of navigation satellites from the United States’s Global Positioning System and two of its counterparts, Europe’s Galileo and Russia’s GLONASS system—81 satellites in all. Most of them have directional antennas transmitting toward Earth’s surface, but their signals also radiate into space. Those signals, say the researchers, are strong enough to be read by spacecraft with fairly compact receivers near the moon. Cheung, Lee and their team calculated that a spacecraft in lunar orbit would be able to “see” between five and 13 satellites’ signals at any given time—enough to accurately determine its position in space to within 200 to 300 meters. In computer simulations, they were able to implement various methods for improving the accuracy substantially from there.
All of the GNSS constellations fit within a 60,000 km sphere close to 400,000 km away from the Moon, which puts them in a 8 degree wide circle. No wonder the resolutions would be hundreds of meters at best, even with higher gain antennas than we use on Earth.
But how can that figure of "within 200 to 300 meters" be estimated? Is there some way to show quantitatively that the same effects that results in several meters of uncertainty on Earth naturally translates to several hundred meters at the distance of the Moon?