In addition to the jets, more than a dozen surveillance aircraft, including armed Reaper and Predator drones, gathered imagery of the events below, while Apache attack aircraft flew sorties in support of ground troops and jammer aircraft sought to block militant communications.

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The result: more than 80 precision munitions fired, destroying four car bombs, four mortar systems and an Islamic State compound, along with other militant targets.

The slew of military aircraft above Mosul that day offers a snapshot of the ongoing air operation that the United States and its allies have mounted in an effort to defeat the Islamic State in one of its final major strongholds.

“Their objectives and us protecting them — that was a good day,” said Air Force Brig. Gen. Matthew Isler, who oversees coalition air and cyber operations for Iraq, referring to support for Iraqi forces.

It also was a noteworthy day, American officials say, because U.S.-supported ground forces took no casualties. Since the beginning of the operation on Oct. 17, Iraq’s military, especially its U.S.-built counterterrorism force, has taken heavy losses, often from suicide bombs or sniper fire, raising questions about Iraq’s ability to sustain a prolonged assault.

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The air operation is a central aspect of a military campaign that has cost $12.5 million a day in Iraq and Syria and that has destroyed countless military vehicles, command centers and fighting positions. Since strikes began in the later summer of 2014, U.S. and allied aircraft have conducted more than 16,000 strikes in both countries. U.S. military officials estimate that 50,000 Islamic State fighters have been killed.

But even such a massive air operation has not been able to extinguish a resilient Islamic State force in and around Mosul and other militant strongholds. As Iraqi forces have pressed more deeply into the city, fighters have inflicted heavy casualties on Iraqi forces and remaining residents, slowing the battle’s progress and damaging the morale of local fighters.

In recent months, U.S. and allied officials have sought to accelerate air operations in the Islamic State campaign, in an attempt to make greater progress against the group, but the pace remains slower than that of other recent conflicts.

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The United States and its allies have flown fewer daily strike sorties in Iraq and Syria than in earlier conflicts, according to Dave Deptula, a retired senior Air Force official who is dean of the Mitchell Institute for Aerospace Studies. In the initial period of the 2003 Iraq War, there were almost 800 strike sorties per day, Deptula said. In the 1999 Kosovo air operation, almost 300 strike sorties were flown a day; in the 1991 Desert Storm operation, more than 1,200.

According to Col. Joe Scrocca, a public affairs official for the U.S.-led operation against the Islamic State, the number of strike sorties had averaged about 56 a day over Iraq and Syria since 2014.

Deptula said the nature of the mission against the Islamic State, and especially the battle for Mosul, requires a different kind of air operation. “Desert Storm was a whole-country operation,” Deptula said. “Mosul is a very small subset of that. … Every conflict situation is different.”

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The Mosul air campaign is being conducted over a densely packed city whose residents had mostly stayed in their homes, making some strikes impossible and increasing the need for precision, low-yield munitions.

“The level of sophistication is an order of magnitude greater” in Mosul than in some past conflicts, said Army Brig. Gen. Scott Efflandt, who heads the U.S.-Iraqi operations center in Irbil, the capital of Iraqi Kurdistan to the east of Mosul. “Here the sophistication of the enemy, the complexity and the scope and size of the city” make for a much more difficult fight, he said.

U.S. officials describe a range of steps that are required to authorize every individual airstrike, including verification of intelligence and a sign-off from military lawyers. Pentagon officials say rarely have they taken a strike that they have known would kill any civilians.

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At the same time, the U.S. military has acknowledged that its air attacks have killed at least 119 civilians in Iraq and Syria since 2014. Watchdog groups say the true figure is probably much higher.

The effort to cripple Islamic State defenses in Mosul shows both the power and limitations of coalition air operations.

While officials have been able to take out car-bomb factories and certain mortar positions, they acknowledge that militants have adapted to U.S. tactics, including the rules on avoiding civilian casualties.

In one recent incident, militants were firing mortars from between closely situated dormitory buildings at a university in Mosul, banking that such a target, between potentially inhabited buildings, would be deemed off-limits to firing. According to U.S. military officials in Iraq, no strike on that target was made.

Neither are U.S. and allied aircraft always able to respond in time to protect Iraqi forces or civilians in the line of fire. That is a particular challenge because, Efflandt said, Islamic State fighters have increasingly embraced attacks on civilians as they face increasing military pressure.

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A little after 5:30 p.m. on Nov. 20, U.S. officials overseeing the air operation received an urgent report from Iraqi counterterrorism troops in the Mosul area. Islamic State fighters had gathered a group of civilians in an open area within the city, and appeared to be preparing to take action against them.

A drone was directed to the site to find out more. Through the drone’s grainy feed, military officials at a nearby headquarters could see the militants behead one civilian, then shoot two more.

The officials immediately requested permission to attack — they hoped to fire not at the militants themselves, who were still gathered around the civilians, but at an area just behind them. Six minutes later, an aircraft was in place and ready to fire. The laser-guided bomb exploded just behind the group, scattering the extremists.

It turned out to be a temporary effect. Less than 20 minutes later, the militants came back, and began to bury the bodies.