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How do spherical harmonics help scientists map Earth’s gravity field and geoid?

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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.

Spherical harmonics are essential mathematical tools used to describe Earth’s gravitational field and the geoid because they provide a way to express variations in potential over a nearly spherical body to any desired degree of accuracy.1

Because Earth’s gravitational potential satisfies Laplace’s equation in the region outside the planet, it can be represented as a series of spherical harmonics. These harmonics consist of terms involving Legendre polynomials that account for mass distribution irregularities, such as Earth’s equatorial bulge or pear-shaped deformations.12

Scientists map these features by:
* Analyzing Satellite Orbits: As satellites orbit Earth, they experience perturbations—deviations from simple Keplerian ellipses—caused by these gravitational irregularities. By tracking these orbital changes, researchers determine the coefficients (dimensionless numbers) that define the importance of different harmonic terms.12
* Synthesizing Data: A global map of the geoid is created by synthesizing these spherical harmonics. There is a direct mathematical relationship between these harmonic terms, gravity anomalies, and geoid undulations, allowing scientists to calculate the geoid's shape by summing the known harmonic components.1