I must be missing something from the story. Several bits confuse me, and I do not understand the point of the question:
The fog is thick enough that the person can't find his/her own way to dry land. The person in trouble has a mobile phone, fine, so at least they can ask for help. The aircraft has a GPS, of course. But all of this gives no clue as to how you are supposed to find the lost person. It is foggy, so you can't see him. Decent fog is a pretty good blanket to TI images (on the frequencies used by police units, anyway), so the infra-red won't help.
We can assume that the fog is sea fog, and that you could fly to and from the police base visually, and fly above the blanket. It still doesn't help find the person................