Yes. While you couldn't target the Field with its original activation, since you choose targets before you pay costs, you can target it with the copy, since the copy is put on the stack only when Rings' triggered ability resolves, well after the Field has been sacrificed.
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
Yes, assuming you own Petrified Field. Because you sacrificed Petrified Field as a cost when you activated its second ability, it will be in the graveyard by the time you choose new targets for that ability's copy.
This goes for petrified field, am I able to target the field with the second trigger?
Might consider getting it now.