Henry Sands has expressed his "deepest gratitude" to rescue volunteers for their efforts in trying to find his father Julian Sands.
The Hollywood star, 65, had been hiking the Mount Baldy trail in the California mountains when he was reported missing on January 13.
Search and rescue crews have been looking for the actor since he disappeared, but so far have found no sign of him.
Their efforts have also been hugely affected by adverse weather conditions.
Speaking about the search, Henry said: "There is nothing really further to say at this point other than to reinforce my deepest gratitude for the love and support my family and I have received.
The 37-year-old said he is being "realistic" about the impact of the bad weather following a 40-year snowfall record in California.
"I am hugely appreciative for all the efforts made so far from the volunteer search and rescue climbers and the San Bernardino county sheriff team to bring my father home.
"I look forward to the search resuming as soon as the weather improves and the ground conditions are safe and feasible to do so," he told The Times.
Five weeks after Julian went missing, the search recommenced.
The San Bernardino county sheriff's department, which is conducting the search, has been using drones in an attempt to find the missing actor.
The initial search for Julian led crews to discover a car believed to belong to the actor.
Phone data showed the last ping from his mobile occurred two days after his disappearance, suggesting that his phone must have died shortly after.
Actor and Julian's hiking partner Kevin Ryan said he believed "something has gone wrong" after the actor's disappearance.
He said: "For me, each day that does go by puts more pressure and I guess brings a lot of doubt in.
"But Julian is such an advanced hiker. It’s what he did. His whole life he was climbing mountains. It was a true passion of his.
"He is beyond experienced and I'm hoping that he's tucked away somewhere and can pull through."
Julian Sands is best known for his role in A Room With A View, where he starred opposite Helena Bonham Carter, Sir Daniel Day-Lewis, Dame Judi Dench and Dame Maggie Smith.
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