The lower mantle structure is dominated by a pair of large, antipodal, low shear velocity provinces (LLSVPs) located beneath Africa and the Pacific Ocean. Though the LLSVPs are a dominantly long-wavelength (degree 2) feature detected since the earliest tomographic models, their nature and origin remain enigmatic. A number of hypotheses have been proposed to address their origin, summarized by two end-member scenarios: (1) they represent thermochemical piles that are either primordial or have grown over time, such as by the accumulation of subducted oceanic lithosphere, (2) they are purely thermal features, seen through the lens of tomographic imaging. In order to distinguish between these two interpretations of the seismically detected LLSVPs we compare the amplitude and length-scales of velocity heterogeneities within and outside the LLSVPs, and analyze their variation with height above the core-mantle boundary. This requires estimating the wavenumber spectrum of heterogeneity by localizing it from a global tomographic model. Previous researchers have done this by filtering using spatially abrupt windowing functions; however, this procedure leads to unreliable spectral estimates due to their non-compact spatiospectral concentration. Here, we overcome this shortcoming by adopting spherical Slepian analysis that allows us to optimize the trade-off between spatial localization and spectral leakage. We conduct a quantitative analysis of the similarities and differences in the spectrum of heterogeneity across a collection of global tomographic models allowing us to identify robust features that need to be explained by purely thermal or thermochemical geodynamic models, potentially discriminating between these two scenarios.