Before I go too much further I should explain why bighorn have these rings. The horns on a bighorn ram, and other horned species (antlers are shed each year, horns are permanent), grow until the animal dies. The core beneath the hard sheath is alive with blood running through it. Although the growth will slow as the animal gets older, it will continue and later in life the horns will get more "mass" or thickness and a large ram becomes more obvious. Bighorn, like almost all wild animals, go through a rough period each year as well. For the Rocky Mountain Bighorn above it would be the winter, for the Desert Ram at the top it would be mid to late summer. During this period the animal is nutritionally stressed and the horn growth really slows, leaving a ring which you can see later. Ewes (females) do the same but their horns are so short it's impossible to really distinguish a horn ring from the normal texture you can note above.
What has been studied and is used as a general guide is the deepest ring is the 4th year. If you use that method on mid to old rams you don't have to worry about how many years have been broomed off. Just start with 4 and count to the base as I did on the 2 below.
I've put a series below in the gallery and see if you agree with me. Get some practice before you go out shooting bighorn again.