Imaging the Earth using Green ’ s theorem 1

The Earth is a big place: its radius is about 6400 km. In comparison, the deepest boreholes drilled are about 10 km deep. We thus have little opportunity to take direct measurements or samples inside the Earth: it is mostly inaccessible. And even for the upper 10 km that we can sample, the cost of drilling deep boreholes is very high. This means that inferences about the Earth’s interior are largely based on physical and chemical measurements taken at the Earth’s surface, or even from space. Investigating the inside of the Earth thus resembles that classical black-box problem: determine the contents of a closed box when you can do anything except open the box. If one knew physical fields, such as the gravitational field or the elastic wave field, inside the Earth, one could infer the local properties of the Earth by inserting the measured field into the equation that governs that field, and extract the physical parameters, such as the mass density, from the field equation. However, the fields are measured at the Earth’s surface or sometimes even above that surface. One thus needs a recipe for propagating the measured field from its surface of observation into the Earth’s interior. This is a problem where mathematics comes to the rescue in the form of Green’s theorem. It relates measurements taken at a surface bounding a volume to the fields inside that volume. This principle is called downward continuation. In the following we apply Green’s theorem to a large class of physical systems and show that this theorem only relates measurements at a surface to measurements in the interior when the equations are the same regardless of whether one