CAIRO, Egypt (AP) _ It's hard to defeat a group of extremists who can mingle among civilian supporters and are pros at propaganda. Israel's military faces the same conundrum the United States has encountered elsewhere _ finding that airstrikes are costly in civilian deaths and public support, while ground attacks are risky for soldiers.

That does not mean Hezbollah is winning militarily. But the guerrilla group has so far avoided a knockout by Israel, even as international pressure for a cease-fire has grown. And in the war of perceptions, Hezbollah has only to look strong against Israel and make Israel look bad to win across much of the Arab world, many analysts say.

That was brought into stark focus Sunday when an Israeli airstrike flattened a house in southern Lebanon, killing at least 56 people, mostly women and children. Israel apologized for the deaths and blamed Hezbollah, accusing it of using civilians as human shields.

But the backlash against Israel and its ally America was swift: Lebanese officials reacted in fury and Beirut protesters attacked a U.N. building and burned American flags. At an emergency U.N. Security Council meeting, Secretary-General Kofi Annan said he was ``deeply dismayed'' his previous calls for a cease-fire had been ignored.

The United States knows this scenario well from Iraq and elsewhere: Pictures of dead children and women killed in airstrikes can hurt support even among friends.

Yet the alternative for Israel, if it wants to push back Hezbollah, is either a full-scale ground war or a lengthy series of smaller-scale incursions to eliminate the group's positions along the Israeli-Lebanese border.

For now, Israel says it has no plans for a big land invasion, still leery from its costly occupation of south Lebanon from 1982 to 2000. But the smaller incursions have brought relatively high Israeli casualties and low apparent impact: U.N. observers in south Lebanon say Hezbollah's supply of rockets remains adequate to fight, and most of its leaders have survived.

Israel has privately told the United States it needs 10 days to two weeks to accomplish what it wants.

Hezbollah's strength comes from its ability to hide fighters and weapons _ both among the populace and in bunkers and tunnels _ who can pop up once the Israelis pass by and fire more missiles toward Israel. That ability springs from its wide support among people in southern Lebanon.

Defense Secretary Donald H. Rumsfeld famously called it ``asymmetric warfare'' and identified it as the challenge America faced from terror groups after the Sept. 11 attacks, and from al-Qaida linked groups in Iraq.

Israel faces just such a struggle against both Hezbollah in Lebanon and the militant group Hamas in Gaza, says Jon Alterman, a Mideast expert at the Center for Strategic and International Studies in Washington.

In many ways, such threats ``are more difficult to resolve'' than battles against conventional military forces, he said. ``The groups have made a living out of having few tangible assets to attack. In many ways, they exist principally as a set of ideas ... and they enjoy wide support among their target communities.''

Israel, of course, has years of experience fighting the guerrilla-style Palestinian uprisings in the West Bank and Gaza. But its wars against outsiders have mostly, except in Lebanon, been against Arab countries' armies or air forces.

Some analysts say that history appears to have left it off-balance this time.

``It's relying too much on the air campaign and it's wrong,'' said Efraim Inbar, an analyst at Bar-Ilan University in Israel.

He instead advocates a robust ground attack and attacks on Syria to prevent Hezbollah resupply of weapons.

But ground attacks also carry risks: Israel lost nine soldiers in ambushes Wednesday alone in operations around the Hezbollah stronghold of Bint Jbail.

Even when Israel succeeds in such pin-pointed ground incursions, ``Hezbollah can disperse, hide men and equipment'' and live to fight another day, notes Anthony Cordesman, another Mideast expert.

And a longer-term occupation of south Lebanon would simply give Hezbollah a ``new, exposed ambush zone,'' plus ample opportunity to raise anti-Israeli and anti-American hostility among Arabs _ a propaganda ploy it is expert at.

Even one of the best outcomes for Israel _ the insertion of an international force at the border to keep Hezbollah at bay _ comes with huge risks for whoever makes up the force, eerily resonant of the attacks against U.S.-led coalition forces in Iraq, Cordesman warns.

``The international force will probably have to do the heavy lifting, be willing to fight and become the focus of new Hezbollah attacks and ambushes,'' he says. ``Non-Muslims will be seen as occupiers and crusaders ... Can anyone spell IED?''

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Sally Buzbee is the AP's Chief of Middle East News, based in Cairo.