Mathematical and Statistical Analysis of Symmetry-Breaking Spectra in Raster Scalar Fields: A Reproducible Framework Applied to Digital Elevation Models of Volcanic Cones

Introduction: We develop a reproducible mathematical and statistical framework for quantifying symmetry breaking in discretised raster scalar fields, with digital elevation models of volcanic cones used as controlled topographic applications. The aim is to formalise, compute and test symmetry-breaking spectra on real elevation surfaces.

Methods — Shuttle Radar Topography Mission 30 m digital elevation models, extracted through Google Earth Engine, were used for three Virunga volcanic cones: Nyiragongo, Mikeno and Karisimbi. The workflow was implemented in Python using NumPy, SciPy, Rasterio, Pandas and Matplotlib. Each raster was reprojected to UTM zone 35S, centred on a morphologically defined summit or crater, cropped to a square domain and normalised as a scalar field. Four finite transformation groups were applied: the cyclic rotation group of order two (C2), the cyclic rotation group of order four (C4), the four-element dihedral group (D2) and the eight-element dihedral group of the square (D4). For each group, an orbit-averaged invariant projection was computed, and symmetry breaking was measured as the ratio between quadratic residual energy and total field energy. Global spectra, sliding-window spectra, multi-resolution tests and Monte Carlo phase-randomised null models were derived.

Results: For Nyiragongo, the global breaking spectrum was C2 = 0.1327, D2 = 0.1880, C4 = 0.2020 and D4 = 0.2056. These scores were below the corresponding phase-null means, from 0.5060 to 0.7677, with p-values from 0.0067 to 0.0133. Local analysis also ranked C2 as the least-broken group, with mean scores decreasing from 0.9022 at 1.92 km to 0.6829 at 7.68 km. Mikeno confirmed transferability, with C2 = 0.1048 and p = 0.025. Karisimbi produced a contrasting case, with C2 = 0.2926 and p = 0.140.

Conclusions: The pipeline provides a reproducible mathematical/statistical framework for estimating, mapping and testing symmetry-breaking spectra in volcanic-cone topography.