The snowstorm that swept into the state early this morning, dropping a foot of snow in some areas, is beginning to ease up, but people should still be careful during the afternoon commute, the National Weather Service says.

With the heavy snow offshore as of early this afternoon, the intensity of precipitation will be much lighter in the coming hours, but “light wintry precipitation will continue and result in slippery travel through this evening,” the forecasters advised in a discussion posted on the Internet.

The snow will linger longest across Northern Massachusetts into Southern New Hampshire, where an additional 1 to 3 inches of snow are possible. In other areas, a wintry mix of snow, sleet, and freezing rain will fall into the evening. Along the state’s southern coast and on the Cape and islands, the precipitation will be all rain, the weather service said.

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With temperatures dropping below freezing in the greater Boston area, refreezing of standing water is also likely in greater Boston and the northeastern portion of the state, the weather service said.

As of about 4 p.m., 9.7 inches of snow had been recorded at Logan International Airport in Boston. The highest total in Boston was 11.0 inches, recorded in West Roxbury. The highest snow totals in the state were recorded in Douglas (12.5 inches), Lunenburg (12.2 inches), and Uxbridge (12.1 inches).

Shortly before 4:30 p.m., weather service radar showed the weather clearing in Western Massachusetts as the storm marched northeast, out to sea.

The one-day storm, which arrived in the early morning hours and dumped up to 2 inches per hour in some areas, blanketed roads and sent some people spinning out. But many appeared to have heeded officials’ advice to stay home.

Thousands of snowplows cleared the roads statewide and in Boston in response to the storm. Governor Deval Patrick ordered “non-emergency” workers to stay home while dozens of school systems closed, including Boston’s.

State Police Lieutenant Daniel Richard said this morning that troopers were responding to “a bunch” of car crashes all over the state, but none had caused any serious injury.

“The entire Commonwealth is affected negatively by the snow,” said Richard, who said he did not know exactly how many crashes there were this morning, but that many were related directly to the snow and road conditions.

Speeds on the Massachusetts Turnpike were lowered to 40 miles per hour from the New York border to Boston.

The state Department of Transportation reported it had 2,706 snowplows operating on state roads across Massachusetts this afternoon.

By late afternoon, more than 500 pieces of equipment were still clearing streets in Boston, Mayor Martin J. Walsh’s office said. The mayor lifted a snow emergency and parking ban as of 5 p.m. The office also said the city planned to post police officers at 100 intersections during the evening commute.

Walsh, dressed in slacks and a zip-up sweater, stopped by the city’s help line office this afternoon, where phone operators had taken some 1,800 calls from concerned residents since Tuesday night.

Walsh said the city was responding well to the storm.

“I haven’t had any complaints really about the neighborhoods not being plowed,’’ the mayor said to a swarm of reporters on the eighth floor of City Hall. “The problem is that we still have snowing going on. … So we have to get ahead of that and keep the streets plowed.”

The mayor also urged residents to be careful both walking and driving the slippery streets.

With another storm possible this weekend, Walsh asked residents to help out their neighbors by shoveling crosswalks, ramps for people with disabilities, and around fire hydrants.

Shortly before 6 a.m., the state’s Trial Court announced that all courthouses would be closed today. Boston City Hall, however, was expected to operate on a normal schedule.

Logan International Airport “is open and flights are arriving and departing — slow and steady,” airport officials said in a tweet this afternooon.

Delays and cancellations were reported this morning on some MBTA commuter rail lines, and buses were deployed on the Red Line’s Ashmont trolley line, according to the T’s service alert website. Multiple bus routes were delayed or changed to snow routes this morning, the T reported.

Weather service meteorologist Matt Doody said the storm was bringing heavy wet snow, in contrast to the puffy powder generated by earlier storms this season.

“It’s not like some of the storms we’ve already had where you could quite literally pretty much blow it off your car,’’ Doody said.

Doody also said temperatures will remain below freezing after the storm passes, but they won’t drop into the arctic cold the region has confronted in recent weeks.

The end of the workweek will be dry and sunny, with highs in the upper 20s through Saturday. Some additional snow could fall between Sunday night and Monday, but those forecasts aren’t definite.

“There’s too much muddled discrepancy to give even a reasonable prediction,” said weather service meteorologist Alan Dunham.