Strike conducted within Saudi territory; at least 40 rebels surrender during air strike.
Saudi Arabia said on Friday it will keep up its offensive against Yemeni rebels who attacked across the border until it has cleared them from its territory.
A Saudi official said on Thursday Riyadh had launched air strikes on rebels in northern Yemen after the Shi'ite insurgents' cross-border raid this week.
But the Saudi Press Agency (SPA) said on Friday the strikes were "focused on infiltrators in Jabal Dukhan and other targets within the range of operations within Saudi territory".
Saudi Arabia, the world's top oil exporter, has become increasingly anxious about instability in Yemen, which is facing a Shi'ite insurgency in the north, separatist sentiment in the south and a growing threat from resurgent al Qaeda fighters.
"The entry of the gunmen to Saudi territory, the aggression against border patrols...and presence on Saudi soil is a violation of sovereignty that gives the kingdom every right to take all measures to end this illegitimate presence," SPA reported, citing an official source.
"The operations will continue until all sites within Saudi territory are cleansed of any hostile element."
Riyadh will take unspecified measures to prevent any future incursion by the Yemeni rebels, the source said, adding that armed forces units have been deployed to back border guards.
The rebels accused Saudi Arabia of attacking villages within Yemen.
"Saudi aircraft are bombing the Malaheet province and the surrounding Yemeni border villages," they said in a statement.
On Thursday, Saudi government officials said the air force had bombed Yemeni rebels who had seized a border area inside the kingdom, which they said had been recaptured. The officials said at least 40 rebels had been killed in the fighting.
Yemen's government -- which has long dismissed accusations by rebels that it has colluded with Riyadh to combat them -- has denied that Saudi planes had struck across the border.
At least 40 of the rebels surrendered during the air strike in the Jebel Dukhan region, an Al Arabiya correspondent told the Saudi-owned network from the region, citing unnamed sources.
Saudi aircraft and artillery were still pounding targets, the reporter said.
Saudi Arabia said on Wednesday a security officer was killed and 11 were wounded in an attack by gunmen who had crossed the border from Yemen -- the first such reported incursion since the long-running Houthi revolt flared up again in August.
The Shi'ite rebels, known as Houthis after the family of their leader, have previously accused Saudi Arabia of backing Yemen's armed forces in the conflict. Sanaa had denied this.
The 1,500 km (930 miles) border with Yemen is a security worry for Riyadh, which is building a high-tech border fence to prevent infiltration.
U.S.-allied Arab countries such as mainly Sunni Muslim Saudi Arabia and Egypt fear Shi'ite power Iran could gain influence in Yemen through the Houthis. The rebels deny getting any help from Tehran, which has offered to mediate in the conflict.
Yemen's army launched Operation Scorched Earth in August to crush the rebellion. Aid groups say around 150,000 people have been displaced by the fighting, which first broke out in 2004.
The United Nations refugee agency official in Geneva said on Friday the body hoped an aid convoy carrying its shelter supplies from the Saudi side would be able to enter northern Yemen in coming days, he said. It was not yet clear if the fighting would delay the convoy.
A spokeswoman of the U.N. huminatarian agency said the condition of Yemenis displaced by war between the rebels and government forces "is becoming increasingly challenging".