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Why was Stokes’s formula difficult to use before artificial satellites were launched?

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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.

Before the launch of artificial satellites, Stokes’s formula was the primary method for determining the form of the geoid, but its application was hindered by significant practical difficulties.

The formula requires an integral that extends over the entire surface of the Earth, incorporating gravity anomaly data convolved with a function of angular distance from the point being measured. This function converges very slowly with distance, meaning that to calculate the geoid's undulation at any specific location, one requires comprehensive gravity data from across the globe.

Because large regions of the Earth remained unsurveyed, scientists faced persistent uncertainties when attempting to compute the integral. While researchers tried to extrapolate gravity anomalies into these unknown areas using assumptions of isostatic equilibrium, these methods were inherently limited. Consequently, the reliance on ground-based measurements made it impossible to achieve a complete or highly accurate global model of the geoid until the advent of satellite geodesy, which allowed for a more effective spherical harmonic expansion of the gravitational potential.1