Just as the dove was went from the hands of Noah to hover over the waters of a new creation, the Holy Spirit is the first person we encounter as we seek forgiveness of sins and request to enter into a covenant of eternal life with Christ.
"And when He has come, He will convict the world of sin, and of righteousness, and of judgement: of sin, becasue they do not believe in Me; of righteousness, because I go to My Father and you see Me no more; of judgment, because the ruler of this world is judged." John 16:8-11
I find it interesting that the dove brought back an olive branch! The dove represents the Holy Spirit, and the olive leaf the anointing flowing through the spoken word He gives.