[{ss_characteristics:[{name:"test 1"},{name:"test 2",new_field:"No"}]},{ss_characteristics:[{name:"test 3",test_field:""},{name:"test 4"}]},{ss_characteristics:[]}]

db.collection.update({},{$set:{"ss_characteristics.$[elem].new_field":""}},{multi:true,arrayFilters:[{"elem.new_field":{$exists:false}}]})