Britannica Sources used in this answer

Why do higher-order spherical harmonics show smaller-scale features of Earth’s gravity field?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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 the study of Earth’s gravity field, spherical harmonics are used to represent the gravitational potential as a series of mathematical terms. These terms describe variations in the field based on the planet's mass distribution.1

The relationship between the order of these harmonics and the scale of the features they represent is inverse:

Low-order harmonics correspond to large-scale, global features of the Earth, such as the equatorial bulge (represented by the $J_{2}$ term) or broad, pear-shaped deformations ( $J_{3}$ ).1
Higher-order harmonics incorporate more complex trigonometric functions, which allow the mathematical model to account for increasingly smaller and more localized variations in the gravitational field.1

While these higher-order terms capture more detail than the primary global shape, they are still primarily suited for large-to-regional scale variations. For very localized features—such as those caused by salt domes, faults, or specific mountain ranges—geologists typically rely on gravimeters rather than spherical harmonic expansions, as the latter become inefficient for modeling superficial, small-scale structures.23