(CNN) -- A few seconds could have made all the difference in the fate of a family who spent three days lost in the snowy California woods, according to the helicopter pilot who found them.
Josh and Lexi Dominguez exit a helicopter Wednesday, shortly after being found.
The father of the family, Frederick Dominguez, came running out of the culvert where they had sought shelter when family members heard the sound of the California Highway Patrol helicopter Wednesday afternoon.
"Had he not been moving, we would not have seen him, because the tree line was very dense and he came climbing out of the culvert," helicopter pilot Steve Ward told CNN on Thursday. "We were just very lucky."
Dominguez had arranged branches to spell the word "help" near the culvert, but rescuers didn't see that until they were turning the helicopter around after spotting Dominguez.
The helicopter was on its way out of the area at that point, trying to get ahead of bad weather.
"The small window of opportunity we had to find them, it was nothing short of a miracle," paramedic David White, who was riding with Ward, told CNN on Thursday. Watch White and Ward describe the rescue »
Dominguez and his three children had been lost in the snow since Sunday, when they set out to cut down a Christmas tree.
They sought shelter first in a lean-to they made of branches, then in the culvert under a road. See family's photos of their ordeal »
Dominguez said his daughter Lexi, 14, was the first to hear the helicopter overhead. He said he ran though several feet of snow barefooted to wave it down.
"When they turned around, man, I was just praising God and saying, 'Thank you, Lord. Thank you, Lord,' because I knew we had made it," he said.
Thursday, Lexi was hospitalized after complaining that her feet hurt. Trying to protect their cold, wet feet from frostbite was an ongoing problem while the four were lost in the forest.
Through their three-day ordeal, the oldest son, Chris, 18, tried to keep his younger brother and sister optimistic.
"I didn't want them to really lose hope," he told CNN's Anderson Cooper on Wednesday night, hours after the family was rescued. "Whenever they would freak out, I would just be like, it's all right. We're going to make it through this. This is nothing. Like, we have already been here a couple of days. What's a couple more days?" Watch the family talk about their "scary" ordeal »
But it was hard to stay hopeful lost in rugged terrain, in heavy snow, with no food and few warm clothes.
"I just remember walking and walking and being like, we're not going to make it," said Lexi. "They can't even see us through all this fog. And I just -- there was just a couple of times where I was really, really scared."
"I didn't think we were going to make it," said Josh, 12.
They removed their sodden socks in an effort to stay warm and dry while they waited for rescue, according to the rescuers who found them. They warmed their feet inside each other's shirts to help stave off frostbite.
"You just go to survival mode," Frederick Dominguez said. "Every parent would do that. You would do anything, sacrifice yourself, because these are your kids." Watch mother as she finds out family is safe »
While the family huddled in the culvert, Lexi led them in impromptu singalongs.
"Someone would say, sing this song, and I would be like, OK," she recalled.
"We would all help her, too," her brother Josh added.
Chris told CNN one of the things he would remember most about the experience was "Lexi in there, in the tunnel, singing her heart out."
The four were reported missing Monday night by Dominguez's former wife and the children's mother, Lisa Sams, according to police in Paradise, California, a town of 27,000 people north of Sacramento.
Although police found the family's car, it offered no clues as to where they might have gone. The remote area is beyond the reach of cell phones, authorities have said.
More than 80 searchers scoured the woods Wednesday until the four were found about 1 p.m.
White said the family was found north of where ground crews were searching. All four appeared to be in good condition as they were brought by chopper to the command post and taken to ambulances.
After an emotional reunion with her mother Wednesday night, Lexi felt pain, said Brian Clarke, her mother's fiancee.
"She woke up in the middle of the night and her feet were hurting her really bad, so I carried her out to the van and Lisa took her to the hospital," he told CNN. "She just kept saying, 'I'm sorry, I'm sorry.' I told her she didn't have to be sorry, we just wanted her to get better."
Her brothers have not complained of health problems that require hospitalization, he said.
However, doctors asked the other three family members to come in for checkups, said Barbara Mejia, the girlfriend of Frederick Dominguez.
Lexi was at Feather River Hospital in Paradise, California, on Thursday, her father said.
"I'm glad I'm home. Praise God," Dominguez told reporters after exiting a chopper at the search command post. "It was awful."
Asked how he survived, he replied, "Jesus Christ."
Butte County Search and Rescue dispatcher Madde Watts said, "They had angels with them, for sure."
Mayor Alan White said he and many others in and around Paradise have cut Christmas trees in the same place where the family vanished. When winter weather sours there, he said, people in the woods can get lost quickly.
"If you're 50 feet from your car, you might not be able to find it," he said.
"We weren't prepared at all," Chris said. "We just thought we were going to go up to the mountains, get our tree and go back home. It didn't turn out that way." E-mail to a friend
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