Hmm. I think we need to move away from "detection ranges" all together. They're quite an artificial contruct, and don't really resemble reality imho.
For example, at range, at night, one factor that will get a U-boat spotted is a larger than usual bow-spray, but whether that results in it being spotted is conditional on many other factors, one of which is the field of view of the searcher through his binoculars, their magnification, and whether or not he's looking in that precise direction when the larger than usual bow-spray occurs.
Rather than posit some mathematical formulae for calculating this, what I'm suggesting is a situation whereby there is no "safe" range for a U-boat able to see the convoy, and not be seen, and therefore no certainty on the part of the U-boat commander that he's unseen because he's "outside visual range".
instead, one could have a chance per second of being seen by an escort, which starts very very low indeed and slowly increases with reduction in range, the change in chance increasing exponentially as the U-boat becomes close to the convoy. The same mechanism could be used when calculating whether - or not - a periscope is seen, leading to the desirable situation where long periods of having the periscope up become dangerous even if submerged. Once spotted, an escort will find it easier much easier to re-acquire a U-boat in the same area if visual contact is lost in the interim.
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