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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?

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    $\begingroup$ The answer to the last question is a definite "no". There's no atmosphere and no surrounding landscape that causes reflections in Moon orbit. $\endgroup$
    – asdfex
    Mar 31 '20 at 10:42
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    $\begingroup$ Is there a "from" missing in the title? ..... that receiving GNSS signals "from" Earth .... $\endgroup$
    – Uwe
    Mar 31 '20 at 11:16
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    $\begingroup$ @Ludo Part of it may be related to: "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." $\endgroup$
    – uhoh
    Mar 31 '20 at 12:48
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    $\begingroup$ @uhoh GNSS positioning is a least-square fit, so if you have your satellites far away with small angular separation, you'd get a poor-defined optimisation problem and thus fairly high inaccuracies. Not sure how tractable the math is for a back-of-the-envelope estimate. $\endgroup$
    – Ludo
    Mar 31 '20 at 13:03
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    $\begingroup$ @Ludo - Re aside from not having to worry about ionospheric/atmospheric effects? Those effects do exist and are significant. The GNSS signals that can be received near the Moon are from those satellites that are just about to go behind the Earth and those that have just appeared from behind the Earth. The GNSS satellites behind the Earth obviously are not in view, and while satellites between the Earth and Moon are in view, their antennae are pointing in the wrong direction. $\endgroup$ Mar 31 '20 at 13:23
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Using Dilution of Precision (DOP)!

I am ignoring the leakage and line-of-sight problems discussed in the comments and just focusing on the geometry. I created a simulation with 4 GNSS satellites randomly distributed in a 53° inclination around Earth and the Moon randomly distributed in a 22° inclination orbit (relative to Earth equator, taken from this Astronomy SE question).

The DOP calculation is well described on wikipedia and here I am using the positional DOP (PDOP).

PDOP values have no units and are just ratios of positioning error to measurement noise [1]. GNSS modules often give an accuracy specification in meters and the product of this specification and the PDOP give the positional accuracy. Importantly, PDOP is independent of the quality of GNSS receiver used, it is only dealing with geometry. The wikipedia article gives an interpretation guide that puts a threshold of ~50 on usefulness.

I ran the simulation with 100,000 points and this is what the distribution of PDOPs were:

PDOP plot

Median values of PDOP are typically in the low 2000's which puts the required terrestrial accuracy specification at around 10 cm. This doesn't seem too far fetched, and is routinely achieved on Earth (albeit with augmentations like base stations), from GPS.gov:

High-end users boost GPS accuracy with dual-frequency receivers and/or augmentation systems. These can enable real-time positioning within a few centimeters, and long-term measurements at the millimeter level.

(emphasis added)

In conclusion:

The 200-300 meter accuracy can be estimated by finding the typical (median) PDOP at lunar distances and multiplying by a specified positional terrestrial accuracy of ~10 cm.

1: Thompson, Ryan & Balaei, Asghar & Dempster, Andrew. (2009). Dilution of precision for GNSS interference localisation systems.

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    $\begingroup$ Thanks for your answer! When you say "I ran the simulation with 100,000 points..." can you mention exactly what simulation you ran? Did it take into account the GPS satellites' antenna radiation patterns that are primarily focused only on the Earth, but for which the newer ones have side lobes that shoot past the Earth into cis-lunar space when they are behind the Earth and either going into or out of radio occultation? $\endgroup$
    – uhoh
    Aug 5 at 22:16
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    $\begingroup$ @uhoh The simulation only varied the positions of the Moon + satellites in the described orbital shells. I did not account for anything other than where the satellites and the Moon were at point n of 100,000 (even ignored the Earth blocking satellites). $\endgroup$ Aug 5 at 22:23
  • $\begingroup$ You mean it ignored that there's only certain times when the antennas' beam is even pointing towards the Moon and can be heard? (refer to info in Has GPS been used beyond GEO?) I guess that plus Earth occultation would just change the statistics, far fewer number of points would be in the 10³ ~ 10⁴ PDOP range, there'd be a lot more complete failures. Where I'm going here is that your ~10 cm starting value comes from augmentation; is that consistent with the paper I've linked to and based my question on? That's really what I've asked about. $\endgroup$
    – uhoh
    Aug 5 at 22:42
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    $\begingroup$ @uhoh yes (quick & dirty sim). The abstract of the paper suggests that the 200-300 m error does not use augmentation. $\endgroup$ Aug 5 at 23:11

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