How does Prospero defend the sinking of the ship to Miranda?

Prospero explains his motivations for causing the storm by telling Miranda his history with the nobles aboard the ship; he reveals to Miranda that Antonio is his brother, and that he was once the rightful Duke of Milan, a position Antonio now holds. Antonio usurped Prospero's estate and wealth while Prospero became increasingly "rapt in secret studies" and oblivious to his brother's machinations; and in order to take Prospero's title as well, Antonio arranged to have his brother Prospero and Prospero's daughter Miranda killed secretly. But Prospero is widely known to be a good man, so those charged with his death decide not to kill him, Instead, Prospero and Miranda were set adrift on the open sea in a decayed vessel, and were able to survive off the supplies that the honest councilor Gonzalo arranged for them to have; thus, they landed on the island where they now live.