A Reproducible Framework for the Compressibility of Truncated Spherical-harmonic Planetary Potential-field Models

Palaash Gang, Aaren Bhandari, Kianu Anggun

For a planetary potential field whose degree-variance spectrum decays as a convergent power law σ²_ℓ ∼ ℓ^−α (α > 1), a single measured exponent simultaneously fixes three classically distinct rates: the truncation-error decay rate, the Sobolev smoothness index, and the Kolmogorov n-width rate. Spherical-harmonic degree truncation is the standard yet inherently lossy way to compress such fields, and this identification turns its efficiency into a direct readout of spectral character. Our contribution is empirical and information-theoretic: we measure the collapse of all three rates onto one exponent, field by field, across real gravity, geomagnetic, crustal, and planetary data (Earth, Moon, Mars), and we map where the power-law regime holds and where it breaks. The fields are synthesized via exact Gauss–Legendre quadrature and checked for implementation exactness against the Parseval identity to better than 10⁻¹³. Compressibility tracks physical source depth: the deep, dipole-dominated core field proves intrinsically compact, the shallow crustal field remains nearly incompressible, and the geoid occupies the intermediate space. Because spherical-harmonic truncation is coefficient-optimal for these red-spectrum fields in the rate (order) sense, residual storage inefficiency stems from bit allocation rather than basis choice; we quantify it by comparing float32 storage against an independent-Gaussian rate–distortion reference and realized coders. All data, code, and figures in this work reproduce from a single pipeline command.