The 51-year-old's white 1991 Nissan Sunny station wagon was found at the Owhiro Bay car park on the city's south coast on Thursday, August 26.
The last confirmed sighting of him was at Woolworths supermarket in suburban Kilbirnie at 9.30am the day before.
Police search and rescue, police divers and LandSAR volunteers spent hours searching the Owhiro Bay area following his disappearance, but nothing was found.
Friends and his brothers Ian, Andrew, and Jim also helped with the extensive search.
Andrew Logan told NZPA in all likelihood had his brother suffered an accident near the south coast, he could have been found.
"We have reason to hope that he is still alive, which is why were are trying to trace his movements, contact people he knew, who may have information about what he was thinking and doing prior to his disappearance."
He said his brother, a conservationist who had a law practice in Kilbirnie, was a hard worker, clocking in "ridiculous" long hours.
To get rid of stress, he would botanize on the south coast or go plant trees behind Oku Street in the suburb of Island Bay.
His disappearance was out of character and the family was deeply concerned about Robert's welfare, Andrew said.
Robert last spoke to his 85-year-old mother and brother Ian on the Thursday evening he went missing, and "seemed perfectly OK and happy," certainly not "irrational or depressed in anyway".
"It's an awful situation when there is no closure. We have absolutely no idea of what's happened. You imagine all the scenarios. We are not blind to the possibilities of what may have happened," he said.
"If he is alive and if he reads this we want to say, 'Robert we want you back. We want you with us, give you a big hug'."
Andrew pleaded for anybody with information about Robert's whereabouts to contact police.
"Don't hold back. We desperately want to know what happened to Robert.
"We miss him; we want him back."
Senior Sergeant Simon Feltham said police had spoken to a number of people as part of the investigation and urged anyone with any information to contact police.
