At least 15 Palestinians, including three children, were killed in the attacks from air and sea, Palestinian medical officials said.
The military said the open-ended operation aims to deliver a blow against the Islamic militant group, and end the rocket fire that has reached deeper into Israel in recent days.
"It won't end in a day and it won't end in two days. It will take time," Yitzhak Aharonovitch, the country's Cabinet minister for internal security, told Channel 2 TV, during a visit to the rocket-scarred southern city of Ashkelon.
"If we need to go inside in a ground operation, then we will do it. These things are on the table. These options exist. We will not stop anything until the rocket firing ends," he added. Asked whether there were any efforts to reach a cease-fire, Aharonovitch said, "Not now."
Israeli officials said the government had authorized the army to mobilize an additional 40,000 troops, if needed, for the operation. By nightfall, the army said it had mobilized half of the forces, in addition to 1,500 reservists earlier activated.
The rocket attacks and Israeli counterstrikes have intensified in recent weeks as tensions have soared over the killing of three Israeli teenagers and the apparent revenge killing of a Palestinian teenager by three Jewish suspects. Following the kidnappings of the Israeli teens on June 12, Israel launched a massive crackdown on Hamas in the West Bank, leading to the surge in rocket fire from Gaza.
Tuesday's fighting was the heaviest since a similar Israeli offensive in November 2012. Israel's military said 130 rockets were fired into Israel. One barrage set off air-raid sirens in Tel Aviv, some 45 miles (70 kilometers) away, signaling the deepest strike yet. The army said the rocket was intercepted and shot down.
The siren set off panic in Tel Aviv, the country's commercial capital, as people scurried for cover in nearby buildings.
Israel's military said it targeted the homes of several Hamas operatives allegedly involved in rocket fire, militant compounds and concealed rocket launchers. Later, it also took out what it said was a Hamas command center embedded within a civilian building.
Palestinian medical official Ashraf al-Qidra reported at least 15 people dead, including at least five civilians.
Among the dead were six people, including two children, killed in an airstrike that flattened the concrete home of a Hamas leader in the southern town of Khan Younis, Hamas officials said. The blast set off a scene of panic as crowds of people, some of them bloodied, fled the smoldering remains. Screaming Palestinians took away motionless bodies, including what appeared to be the two children.
A separate airstrike on a motorcyclist killed two people, including a young boy who was passing by, al-Qidra said.
Late Tuesday, Israeli troops shot and killed four militants who tried to infiltrate a military base in southern Israel by sea. Lt. Col. Peter Lerner, a military spokesman, said the attackers made it ashore and attacked the base with grenades and assault rifles. An Israeli soldier was lightly wounded.
Lerner said Israeli forces were searching the area for other attackers. Hamas claimed responsibility for the attack.
Israeli troops continued to move toward the border in anticipation of a possible ground invasion, most likely in the coming days.
"We will not tolerate rocket fire on Israel cities, and we are preparing to expand the operation with everything at our disposal to strike Hamas," said Israeli Defense Minister Moshe Yaalon, who announced a special state of emergency in southern Israel.
In Gaza, Abu Obeida, a masked spokesman for Hamas' military wing, accused Israel of violating a cease-fire that ended a 2012 round of fighting. "In the face of this aggression, we affirm the Zionist enemy should not dream of calm and stability," it said.
Palestinian President Mahmoud Abbas called on Israel to halt the airstrikes immediately and appealed for calm.
"The Palestinian leadership is conducting intensive and urgent contacts with regional and international parties to stop the escalation," he said.
Abbas, however, has little influence over a Gaza Strip. Hamas took control of Gaza in 2007, and despite a recent unity deal that ostensibly handed control back to Abbas, the militant group remains the dominant power there. Smaller and more radical forces than Hamas are also involved in rocket fire from Gaza.
The military says Hamas has amassed about 10,000 rockets, including longer-range rockets that can reach central Israel. The military ordered hundreds of thousands of Israelis within a 40-kilometer (25-mile) radius of the Gaza Strip, including Israelis in the major southern city of Beersheba, to stay indoors and near shelters.
The renewed rocket fire from Gaza comes as Hamas is increasingly isolated and under pressure from Egypt, where the new regime ousted its former patron the Muslim Brotherhood.
Israel is also cracking down on the organization's West Bank operation following last month's kidnapping and killing of three Israeli teenagers in the West Bank. Israel blames Hamas for the teens' abduction and is conducting a manhunt for two Hamas-affiliated Palestinians in the West Bank it believes carried out the kidnapping and killing.
Tensions have been high since the three Israeli teenagers were kidnapped June 12 in the West Bank and were later found dead. That was followed by last week's slaying of the Palestinian youth. Six Jewish suspects have been arrested.
Heller reported from Jerusalem.
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