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How did Sir George Gabriel Stokes develop a method for calculating geoid undulations from gravity measurements?

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Diagram showing the deflection of the vertical from the geoid to the ellipsoid or spheroid, with labeled lines for geoid, spheroid, rock surface, ocean, and geoid height in a geology context. — geoid Deflection of the vertical from the geoid to the spheroid.

In 1849, the British physicist and mathematician Sir George Gabriel Stokes solved the problem of relating geoid undulations (N) to gravity anomalies ( $Δ g$ ) in a calculable manner.1

Stokes developed an integral equation for N that required integrating gravity measurements over the entire surface of the Earth. In this formula, the integrand consists of gravity anomalies ( $Δ g$ ) convolved with a function of their angular distance from the station.1

While this method served as the principal way to determine the form of the geoid until the advent of satellite technology in 1957, it presented significant practical challenges. Because the function of angular distance in the integrand converges very slowly, calculating N at a specific point required extensive gravity data from across the globe, including from remote or unsurveyed regions. Consequently, modern approaches have largely shifted toward using spherical harmonic expansions, which combine satellite observations with ground-based gravity measurements.12