I would like to add yet another theory to the many which have been proposed over the years as to the cause of LDEs (long delayed echos). Over the years starting in the late 1920s there have been reports in QST
and other Ham magazines of Hams hearing the last few seconds of their transmission echoed after they stopped sending. I am wondering if something in the atmosphere could be exhibiting a very high index of refraction which would slow the radio waves and cause a few second delay. What brings this to mind is the work performed a few years back by the Lene Vestergaard Hau group at the Roland Institute. This group was able to slow light to less than 40 miles per hour by passing it through a Bose-Einstein condensate, a Bose- Einstein condensate slows light due to its very high index of refraction. It is unlikely that a Bose-Einstein condensate could naturally occur due to the extremely low temperature required. If a region exhibititing a high index of refraction were to send a transmission back to the sender it would have to be U shaped or have a region resembling a reflector at one end to send the signal back through the region to the sender

Very interesting topic. I would think there may be other possible scenarios as well. For instance the distance around the earth is almost 25000 miles. Now if a signal were to be passed all the way around the earth it would have to be reflected several times in an up and down zig zag fashion, thereby greatly increasing the distance the signal would have to travel. This could account for receiving yourself after you un-key. I have heard some echo on people on the HF bands which would tend to make me believe I am receiving the signal from more than one path. However this only accounts for a very slight delay, maybe up to 250 ms or so, not several seconds. I am sure there are other theory's out there too.