“The cluster of photons in your picture shows very clearly that there is something on the lunar surface, of very small size, reflecting a lot more photons than all the rest of the surface.”
This looked quite convincing at a first glance…though I did not fully understand the graph at first. Looking at it somewhat closer and thinking about it I found this comment to be off the mark.
To explain it: The y-axis displays the time gate for each individual laser pulse in nanoseconds. This means: When you send out a pulse you expect the reflections to return at a narrow time gate. The displayed range is about 100ns wide, which is equivalent to about + or -15m runtime (length) difference of light photons. Most of the photons have returned at a narrow band of about 2ns, which indicates that their runtime was about the same (less than 1m length difference).
However, anyone able to do the basic math will know that projecting a beam (cone shaped) to the moon ball (~300000km away, radius ~1700km), where it spreads out to about 2km in diameter and then back to the source on earth will yield no measurable runtime difference (< 1 ns) in individual photons, no matter which point inside of this target area was hit - given that the surface is perfectly flat. So all you have to do is actually find a rather flat area on the moon with no notable craters, hills, rocks and mountains and you should quite precisely get the response shown in the graph.
So, far from being proof for the presence of a laser reflector, this graph is only proof that the surface is reflective and the targeted area is flat. On the contrary, if all those photons would have been reflected by the tiny reflector, the band should be a really fine line and not a band, as shown. So this graph can be considered as proof that no reflector is present (or at least it has no impact).
To prove that the reflectors actually are present you would need to compare the alleged landing sites with a large sample of other areas on the moon (especially flat ones) and conclusively show that the increased reflectivity at the “landing sites” is highly unusual. I’m not aware anyone has made such a comparison.
