David said to Abiathar, I knew on that day, when Doeg the Edomite was there, that he would surely tell Saul: I have occasioned [the death] of all the persons of your father's house.
So he actually crossed over in the area of the Dead Sea, went over to the other side to Moab and there he established his family. Now he knew that Saul's anger against him would ultimately turn against his family.