Some safety departments are set up as independently as possible from the operations. 

This is not a wrong setup per se, as long as top management is supporting this and is willing to accept whatever comes out of it.

But often independence also means rejection of the function by operations. (“Ah! What do they know about safety!”) Better choose to be effective instead. Let operations participate in the investigations. Make sure that any recommendations that follow from an investigation is supported and carried by operations.