In Sogetsu, the gagoh is a name you can choose for yourself; but more often, it is your teacher who gives the name to you. In Japan, the study of ikebana is a lifelong endeavour for most people. Once a teacher has been chosen, students study under that teacher for the rest of their lives, except when their is a special reason to change. The gagoh is normally given in kanji, where one of the characters is taken from the teacher’s own gagoh. Miyako's teacher’s gagoh is “shikou” (紫虹 “purple rainbow”). Miyako's gagoh is “kouto” (虹都 “rainbow capital”). Incidentally, the character "to" (都) can also be read as "miyako" in Japanese!
Again, the gagoh is not compulsory but most people like to have one. The gagoh gives recognition of accomplishment and also a sense of continuity from teacher to student. If you do elect to have a gagoh, this must be duly registered with the Sogetsu Foundation. The gagoh appears in official documents like the Sogetsu Teachers' Association Membership Card.