Amin, if there is a better place for this, feel free to move it or let me know. This is something I felt like writing because I feel like it could help some folks out with less than ideal staves still produce a very nice bow.
So I would like to post here about a few lessons I have learned. This is not going to always be true but I suspect more often than not. Feel free to add or correct if you wish.
Thin ringed osage, when chased to a single growth ring has several problems. One being the obvious, the rings are thin and hard to chase. The other lesser obvious is the tool marks. Once your ring is established, you need to clean up your tooling marks, you know, the little nicks, chatter ( AKA wash boarding ) and lines left behind from your scrapper. The marks if left alone are stress risers. They are all potential breaking points from where the ring is partially violated and invites a splinter to pop.
Assuming you pop a splinter in your thin ringed osage you can always super glue it down and put a wrap of thread around it to fix that splinter. But chances are high ( read, its gonna happen ) that another splinter will pop somewhere else. The first splinter popping is telling you something so listen and you may save your bow. Its telling you the back ring is so thin, it cannot do its job of holding in the tension stresses. It needs help.
There are two ways to do this. One is back the bow with rawhide, sinew, silk, linen, or some other popular backing. These provide durable backings that hold down the splinters and relieve the back of some of its stresses. You may not want to back your bow however. The way for you to go then would be to decrown your bow. It really may not take much. Just a few consistent flat strokes with your scraper to give you a path of about an inch wide, following the waves of your bow should do. You may ( probably ) be able to even remove the splinter completely while doing this. Adjusting the tiller afterwards is an easy task. The bows weight will have dropped a few pounds depending on how much wood was removed. Shortening the bow, recurving the bow slightly ( remember the wood is thin ringed, dont overstress it ) or heat treating it will recover your weight lost.
Of course the best way with this when dealing with thin rings is to back, decrown to begin with. One last option that needs be mentioned is the sapwood. If the sapwood is in good shape and has thicker rings than the heartwood, leave it on there. You can chare a sap ring to give you a better back than your heartwood would, giving you a much better chance of holding a bow instead of your head because a limb just went air born...