Surface energy associated with a free plane of atoms in a crystalline solid is manifestly important for processing and predicting the behavior of polycrystalline materials. Minimization of the surface energy at a triple-junction separating adjacent grains requires satisfaction of a fundamental partial differential equation first posed by Conyers Herring in 1952. Using magnesia (MgO) as a test material, atomic force microscopy is used to gather generally noisy experimental data from equilibrated thermal grooves circumscribing island grains. This dataset is then required to satisfy the Herring equation at many material locations along the thermal grooves, leading to a large and overdetermined system of linear equations. The corresponding inverse problem is then solved using a novel technique that is statistical in nature, multiscale in implementation, and draws on the iterative projection algorithm due to S. Kaczmarz in 1937. The resulting discrete solution represents a st atistically significant representation of the surface energy of MgO as a function of surface orientation.