From what I can gather, your board has a square cross-section of 1.5"x1.5" and the picture you showed us is of the right side of the board. Your question is whether or not you could or should chase a ring, correct? If so, then we really can't say for certain without seeing pics of the back and end grain at the very least.
If the grain is perfectly straight on the back and the growth rings are perfectly parallel to the back, then you could chase a ring, but whether or not you should really comes down to a personal choice. My personal opinion is that it isn't worth the effort. You seem to have a beautiful straight grained face to use as a back. There will be plenty of time for learning to chase a ring later. But, like I said, it's up to you.
If you post some more pics, it will really help us to understand your situation and offer more detailed advice.