This project introduces electrical resistivity tomography, a technique of interest for geophysical imaging, used to produce images of underground features or structures by using electrical current. Specifically, a known electrical current is injected into an object (for example, the earth) and the resulting induced voltages are measured on the surface. From this we attempt to determine the interior electrical resistivity of the object. This is an example of an inverse problem.
In this project we first model how electrical current flows through a conductive medium, which results in a classic boundary value problem in two or three dimensions. In the simplest case of interest this is just Laplace's equation. With this model in hand we then consider the inverse problem of recovering a nonconductive internal circular ``void'' in an otherwise uniformly resistive object, as a simple example of how this imaging methodology works.
In this project we would benifit from exposure to basic Vector Calculus and the Divergence Theorem, at a level typical in a third semester calculus course. Some exposure to boundary value problems or Laplace's equation is also helpful, with allied topics like separation of variables and Fourier series solutions, as well as familiarity with the notions of an electric field and electrical potential that one might see in an introductory physics or vector calculus course.
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