Despite a police operation which involved helicopters, boats and land searches, officers have found nothing.
Inspector Drummond said of the operation: 'She went missing at her home address on Boxing Day at 3am…what we've got is an image of her, which we believe to be her, running from the home address and towards the beach on Avondale Road.
Mr Murphy said: “I have been doing this all my life.
She is wearing…a vest top and shorts, which is unprepared for the weather.'What I can say is that from speaking to the family in great detail, Sophie suffers from severe anxiety and depression which she was being treated for.
There was a level of what I would call preparation prior to running off to what happened.''I think a combination of the preparation, her mental state at the time and comments made may suggest there's a likelihood that unfortunately she may have gone into the sea,' he added.
Of her mental state, Inspector Drummond said: 'We believe she's been talking about it with family before, talking about her anxiety and depression and we think this has maybe been a trigger.''The theory is that if she has gone into the sea, unfortunately, we need to be looking at the tides and how that would impact.'We are treating it as a live missing person's inquiry but we have got to take the consideration that there is likelihood unfortunately that she has gone into the sea,' he added.
Earlier, Miss Smith's stepsister Lauren Shaw, 26, and stepfather led around 15 friends and volunteers on a search of the seafront in Gorleston and around Great Yarmouth.