Why did lord carnarvon ask carter to supervise the work

In 1907, after three hard years for Carter, Lord Carnarvon employed him to supervise Carnarvon's Egyptian excavations in the Valley of the Kings. The intention of Gaston Maspero, who introduced the two, was to ensure that Howard Carter imposed modern archaeological methods and systems of recording.

KV62 in the Valley of the Kings
Carnarvon financed Carter's work in the Valley of the Kings to 1914, but until 1917 excavations and study were interrupted by the First World War. Following the end of the First World War, Carter enthusiastically resumed his work.

After several years of finding little, Lord Carnarvon became dissatisfied with the lack of results, and in 1922 informed Carter that he had one more season of funding to search the Valley of the Kings and find the tomb.