NEW DELHI (Reuters) - India's monsoon rains for the week ended July 1 were 29 percent below normal, improving from a 68 percent shortfall in the previous seven days, the weather office said on Thursday. The June-September annual rains are the main source of irrigation for Indian farms and are crucial for Asia's third-largest economy. Countrywide rainfall was 39.6 mm in the latest week against a normal 56.0 mm, the India Meteorological Department office said. It said all 36 weather zones received rains during the week, indicating a revival after a weak phase during the previous two weeks. The monsoon was normal in 10 areas and surplus in three, while 23 zones recorded deficit to scanty rains. Rainfall during June 1 to July 1 was 92.2 mm, 46 percent below normal. It was better than June 1 to June 24 when the rains were 52.8 mm, 54 percent below normal. Last week, the weather office said total rainfall from the four-month monsoon would be only 93 percent of the long-term average, coming in below normal for the first time in four years. Although the weather office says monsoons would improve in July, some forecasters disagree. "It is going to be dry weather in July," Michael Ferrari, vice president of the U.S. based firm Weather Trends International, told reporters Thursday on the sidelines of a sugar conference in New Delhi. He cited two localised factors hurting the progress of monsoons: the presence of strong pressure over central India, and water in the Indian Ocean had not warmed up sufficiently. "Overall monsoon would be a failure," he said. But the weather office was hopeful of good rains in coming weeks. "We are still hopeful that monsoon rains will catch up in the first fortnight of July and to a large extent make up for the deficiency in June," said L.S. Rathore, a top official at the government's weather office. The monsoon hit the southern coast on May 23, ahead of the normal date of June 1, but displayed a weak phase for about two weeks since June 7.