Halfway to its center above the liquid core, the lowermost part of the Earth's mantle is particularly mysterious and exhibits remarkable yet poorly understood seismological features. The most recent tomographic images or models unveil Large Low Shear Velocity Provinces (LLSVPs), which are crucial to understanding past and present Earth's mantle circulation, chemical composition, and thermal structure. We must sharpen our tomographic models that lack resolution for addressing such fundamental issues, bringing new insights into these mysterious structures. A striking observation is the presence of 'Ultra-Low Velocity Zones' (ULVZ) toward the edges of the LLSVPs, near the root of hot plumes rising through the mantle and manifesting as volcanic activity at the surface. ULVZs are small objects, at most tens of kilometers high, well below the resolution limit of current tomographic models. 3D imaging of these extreme heterogeneities' precise contours is a considerable endeavor and has yet to be attempted. I will discuss recent theoretical and methodological advances that allow us to meet this challenge. 1) Box-Tomography makes it possible to obtain localized high-resolution tomographic images at arbitrary locations in the deep Earth where no seismic sources nor receivers are present. 2) The distributional finite-difference methods allow us to model wave propagation with unprecedented accuracy. I will illustrate the relevance of these two approaches for investing our targets of interest at the core-mantle boundary.
