In deciding on an advance in Iraq, Army commanders justify throwing centuries of military doctrine into the trash heap by citing a couple of fairly new and relatively untested theories–the “Rolling Start” deployment and the “Bypass Criteria.” Both are departures from Colin Powell’s doctrine of massive, overwhelming force.

Rolling Start, in which attacks are launched while additional forces continue to arrive, is based on the assumption that the Iraqi Army was so inferior that the United States could begin the ground campaign even though the Iraqis had a 10-to-1 numerical advantage in ground forces (it may even be more than that, though actual figures on how many combat troops are now inside the country are a bit murky). Bypass Criteria, by which regiment-size combat teams decide to skip intermediary objectives and race ahead to major targets, is based on two assumptions: 1.) that the population in the rear is friendly and that bypassed enemy units are not very dangerous, and 2.) that air power could make up for the lack of manpower when it comes to defending the supply line and to securing any possible line of retreat.

Events of the first week of war challenge all of these assumptions, at least to some extent. The Iraqi Army is inferior, but not as inferior as expected; even the 51st Division, a regular army unit in the Basra area that was considered cannon fodder that would collapse on first contact, is still fighting as a cohesive unit in Basra. In Umm Qasr, a small contingent of Republican Guard held out against a larger force of U.S. Marines for four or five days (assuming coalition reports that Umm Qasr was pacified today prove more accurate than similar claims last Friday, and again on Sunday, Monday, Tuesday and Wednesday).

Throughout Iraq, much of the population is indeed friendly, soldiers are indeed seizing opportunities to surrender, but much larger numbers than expected are not only hostile, they also seem devoted to fighting a guerrilla war in our rear. And as for air power being a replacement for boots on the ground: well, we saw what good that did the soldiers from the 507th Maintenance Company who were killed or taken prisoner in Nasiriya on Sunday.

Along what is now a nearly 300-mile-long supply line (which splits after less than 100 miles into three major spurs), I’ve seen key choke points, such as a bridge or an overpass, where as few as three soldiers are standing guard. I can’t remember another modern instance in which American forces operated with such a long and vulnerable tail.

The picture is not all dreary. The advance is impressive, and U.S. and British forces have taken light casualties so far (although most troops have not yet engaged). Bombing exacts a daily toll that cannot be easy for the Iraqis to sustain. Iraqi attacks have not yet managed to cut the supply lines for more than a few hours at a time. In the course of this week U.S. and British troops will be at the red line around Baghdad; some may already be there.

Much could continue to go wrong, though. Major sandstorms, like one that blew for three days this week, could delay the advance–and this is the sandstorm season. Long-enough delays could push the fight into extreme summer weather, which usually begins by April. The Iraqis may even manage to launch counterattacks. What if they actually flank the advance and attack the Third Infantry Division’s rear? Or cut off a battalion of Marines and cut them up before help comes? Certainly U.S. air power, firepower and superior maneuverability would suppress such a move before it goes too far, but the propaganda bonanza will be tremendous. Last week, U.S. military intelligence reported that thousands of Republican Guard troops were on the move, heading to cut off the Seventh Cavalry and other units far in the front. That report turned out to be false–at least for now.

In the meantime, the Iraqis are really scoring points, especially in the Arab world, with the more ambiguous than expected picture of the Iraqi public’s reaction to the invasion. In the first couple days, American troops were greeted warmly, but the Baathists and other loyalists quickly reasserted themselves when it was clear no one was hanging around to hunt them down. Then the general mood sobered quickly as folks began fearing it might be 1991 all over again, when the United States abandoned the Shiite uprising to its bloody fate. It became abundantly clear to them that Washington didn’t have the troops to pacify their communities, arrest the Baathists, put the Saddamites out of office and so on. By the time I pulled out of southern Iraq early this week, no one was smiling and waving anymore–even had they wanted to.

Meanwhile, the United States continues to put its faith in the prospect of a knockout punch in Baghdad. Decapitate the beast, then worry about holding down its twitching extremities. Politically, especially overseas, the Americans could use a quick victory. In the Vietnam War it took decades to approach the level of opposition this war has aroused, even before it started. That can only worsen. Same with the Iraqi public reaction. In Az-Zubayr, I saw a single accidental shooting incident inflame an entire town that had been cheering only hours before. Not far from Safwan on Wednesday, people accepted food and water from an aid convoy while party hacks led them in provocative chants of “With our blood and our souls, we sacrifice our lives for you, Saddam.”

A few days earlier, the same small community was pulling down and stomping on the posters of Saddam. But troops can’t do much about that; they’re stretched too thin, merely holding the line of supply open. Reinforcements are on the way; the Army seems anxious to step up its deployment of troops to Iraq, bringing the Fourth Infantry Division in ahead of schedule, for instance. And along the way, by the middle of this week, they realized they couldn’t just push through when they were sniped on and ambushed; they had to pause and hunt down the perpetrators and put them out of action.

Tommy Franks is certainly no Custer. But he has a much more daunting job than his Iraqi opponents. To succeed, Franks has to subdue the country, take its capital and eliminate its leader. All Saddam has to do is kill enough Americans, and cause enough embarrassments and setbacks that the war becomes unsustainable politically. That hasn’t happened, but I’m not sure the Americans appreciate the danger that it might.