Someone will probably have more detailed information, but basically those lines are smoke flares that are set off just before the explosion to act like lines on a ruler. The flares are set off at a known distance from the explosion (and from the observers), so that by looking at the photos of the explosion afterwards, one can make a good approximation of the size of the blast.
I have a much better picture of the rocket trails than those shown
in the link to the nuclear weapons archive. This pic was taken at
Maralinga in Australia.
However, I cannot see any way to post the pic to this forum, pity.