Wellington monument on the Blackdown Hills, in Somerset, was built in 1817 but since 2005 it has been fenced off because of falling stone debris.
The National Trust is using ground-penetrating radar on the 174ft (53m) tower to see under its stone cladding.
Ken Evans, from the trust, said the work was "crucial".
Built on one of the highest points of the Blackdown Hills, the landmark was put up as a tribute to the Duke of Wellington's military achievements at the Battle of Waterloo.
But according to the trust, it has been struck by lightning twice in its history and renovating the very tall landmark every 10 to 15 years has been "expensive and unsustainable".
Mr Evans, the trust's building surveyor, said the radar study was one of several being carried out to "understand this unique and somewhat complex monument".
"We have been using wind and movement sensors which have already surprised us by showing that it doesn't flex in the wind quite as much as we expected," he said.
"The ground-penetrating radar seeks to identify voids and gaps in the stonework under the surface but should also tell us more about the materials which were used to build the obelisk."
Data from the detailed survey will also be used to build a computer model of the obelisk and help with a "more effective repair approach".

Six members of Cymdeithas yr Iaith Gymraeg blocked the entrance to the building at Llandudno Junction at about 06:30 GMT.
North Wales Police were called to the protest, which ended at midday.
The Welsh government said it was taking action to promote the language.
The protestors said the action was part of a general Cymdeithas campaign to put pressure on the Labour administration in Wales to act urgently for the Welsh language.
They want the Welsh government to adopt six basic policy changes, including making Welsh-medium education available for all and what they call "fair funding" for the language.
Robin Farrar, chairman of the society, said: "We're here to speak out for the present generation, and those to come, who should be able to live in Welsh.
"There's a real crisis facing the language, but the Welsh government is not acting seriously.
"We're certain that our unique national language can thrive over the years to come with positive campaigning and political will."
A Welsh government spokesperson said: "We are very disappointed that Cymdeithas yr Iaith has decided to take this course of action, especially as we've held regular, and constructive, dialogue with the group.
"Over the past year we have taken positive action to promote the Welsh language, including significant measures such as publishing proposed standards to improve Welsh language services to citizens."
A computer model of one of the world's tallest three-sided obelisks is being made to find out why it is falling apart.
[NEXT_CONCEPT]
Welsh language campaigners have chained themselves to the gates of Welsh government offices in a protest at a "lack of response" to the decline in the number of Welsh speakers.