NEW YORK (CNN) -- A "furious" President Obama has ordered a review of the decision to fly a Boeing 747 frighteningly close to the lower Manhattan skyline, White House press secretary Robert Gibbs said Tuesday.

Witnesses reported seeing a plane circle over the Upper New York Bay near the Statue of Liberty.

The review will focus on "why that decision was made and to ensure that it never happens again," Gibbs said. Jim Messina, a deputy chief of staff, will lead the effort.

On Tuesday, Obama told reporters, "It was a mistake. It was something we found out about along with all of you. And it will not happen again."

Officials said that when Obama first found out about the incident, he was "furious."

The incident has led to a large amount of criticism from New Yorkers and politicians.

After a YouTube video showed panicked New Yorkers scrambling as the plane flew frighteningly close to the lower Manhattan skyline, a former Homeland Security adviser questioned whether the man who approved the flyby should remain in his White House office.

Fran Townsend, who advised President George W. Bush for more than three years, called the move "crass insensitivity" in the wake of 9/11.

"I'd call this felony stupidity. This is probably not the right job for Mr. Caldera to be in if he didn't understand the likely reaction of New Yorkers, of the mayor," Townsend said Tuesday on CNN's "American Morning."

Louis Caldera, director of the White House Military Office, quickly apologized for Monday's incident after the planes prompted workers and residents to evacuate buildings in New York and New Jersey. Watch Townsend slam Caldera »

"Last week, I approved a mission over New York. I take responsibility for that decision," Caldera said. "While federal authorities took the proper steps to notify state and local authorities in New York and New Jersey, it's clear that the mission created confusion and disruption."

The Federal Aviation Administration said the aircraft, which functions as Air Force One when the president is aboard, was taking part in a classified, government-sanctioned photo shoot.

On Tuesday Sen. John McCain wrote a letter to Defense Secretary Robert Gates saying he was "profoundly disturbed" by the flyover "against the backdrop of September 11."

"The supposed mission represents a fundamentally unsound exercise in military judgment and may have constituted an inappropriate use of Department of Defense resources," McCain, a ranking member of the Senate's Armed Services Committee, wrote in the letter.

He requested an investigation into the cost of the flyover.

Air Force officials estimate that the mission and the photo shoot for the 747 and an accompanying F-16 fighter jet cost $328,835. But they said "the hours would have been flown regardless, and the expenses would have been accrued on a different mission."

An Air Force source told CNN on Tuesday that the White House Military Office planned a photo shoot over various Washington monuments next week, but the shoot has now been canceled.

An angry Mayor Michael Bloomberg said it "defies the imagination" that an agency would schedule the photo shoot so near the site of the September 11, 2001, attacks on the World Trade Center.

Sen. Charles Schumer, D-New York, said the FAA's decision to not announce the flyby "borders on being either cruel or very, very stupid."

Witnesses reported seeing the plane circle over the Upper New York Bay near the Statue of Liberty before flying up the Hudson River. Watch the plane fly over Manhattan »

The YouTube video shows dozens of people standing in a parking lot, watching the plane approach. As it nears, they begin to run. Someone unleashes an expletive. "Run, run!" says one person. "Oh my God," cries another.

A 911 call released by the Hudson County Sheriff's Department in New Jersey shows the panic caused by the plane.

"There's a plane falling, there is a big aircraft falling like 9/11," a man says to the operator. "Everybody is running and people are crying and panicking."

Two officials said the White House Military Office was trying to update its file photos of Air Force One.

Bloomberg said he was perturbed. Watch the White House apologize »

"I'm annoyed -- furious is a better word -- that I wasn't told," he said, calling the FAA's decision to withhold details about the flight "ridiculous" and "poor judgment."

"Why the Defense Department wanted to do a photo op right around the site of the World Trade Center defies the imagination," he said. "Had we known, I would have asked them not to."

Linda Garcia-Rose, a social worker who counsels post-traumatic stress disorder patients in an office three blocks from where the World Trade Center stood, called the flight an "absolute travesty." Watch the White House respond to questions about the scare »

"There was no warning. It looked like the plane was about to come into us," she said. "I'm a therapist, and I actually had a panic attack."

Garcia-Rose, who works with nearly two dozen patients ages 15 to 47, said she was inundated with phone calls from patients.

"They're traumatized. They're asking 'How could this happen?' They're nervous. Their anxiety levels are high," she said. iReport.com: Tell us what you think

Garcia-Rose said she is considering filing a class-action suit against the government for sanctioning the plane's unannounced flight.

"I believe the government has done something really wrong," she said.

Capt. Anna Carpenter of Andrews Air Force Base in Maryland said local law enforcement agencies and the FAA had been given notice of the exercise.

New York Police Deputy Commissioner Paul J. Browne said the department had been alerted about the flight "with directives to local authorities not to disclose information about it."

CNN's Mythili Rao and Ed Henry contributed to this report.

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