Did you apply any correction before bed or was it just the levemir?
With nothing else if you are seeing a drop overnight then it seems that your levemir is a little high if anything. I am surprised that you are only taking a single dose a day as typically it doesn't last a full 24 hours (according to
http://iddt.org/about/gm-vs-animal-...imes-of-animal-gm-human-and-analogue-insulins they recon about 14 hours) so perhaps you need to look at splitting your dose. Ideally overnight you shouldn't see too much of a rise or fall, a 4mmol drop overnight looks to me like too much levemir at night, if I had that kind of drop I would be backing off my evening levemir by 2 units but I take 2 doses a day of that.
I would strongly suggest that you contact your DSN and discus with them about splitting your levemir dose since you are apparently dropping overnight and rising during the day, however I suspect you are covering for the lack of basal insulin in the afternoon with your rapid insulin doses instead. Have you completed DAFNE or some other similar course? If not then it might be worth doing a quick search and familiarising yourself with how to keep a diary in that format then go to your DSN with that. Being able to see the splits between your food dose and correction doses as well as the daytime and overnight basal activity should help work things out.