The correct answer to the question is option 4, which is the Vicinity of Thamirabarani river. Archaeological studies have revealed that by 1200 BCE, there was a flourishing agrarian civilization and rice cultivation in this area.

Option 1, Raichur Doab, is not the correct answer because there is no evidence to suggest that there was a flourishing agrarian civilization or well-established rice cultivation in this region by 1200 BCE.

Option 2, Krishna Delta, is also not the correct answer because there is no evidence to support the existence of a flourishing agrarian civilization or well-established rice cultivation in this area by 1200 BCE.

Option 3, Vicinity of Vamsadhara river, is not the correct answer because there is no evidence to indicate the presence of a flourishing agrarian civilization or well-established rice cultivation in this region by 1200 BCE.

The correct answer, option 4, Vicinity of Thamirabarani river, is supported by archaeological studies which have revealed the presence of a flourishing agrarian civilization and well-established rice cultivation in this area by 1200 BCE.