Inverse method for imaging a free surface using electrical impedance tomography

The feasibility of inverting measurements from electrical impedance tomography (EIT) to image the free surface of a homogeneous conductor within an open channel is investigated. Potential measurements from a linear array of electrodes placed at the bottom of the channel are simulated using a "nite element method. The simulated measurements are inverted using an iterative, sometimes weighted, least-squares method to give an approximation to the height of the conductor as a function of position. The inverse problem is not well-posed and is also ill-conditioned. To pose the problem properly the free surface is either described by a global parameterization or a local, piecewise parameterization, where in both cases the number of coe$cients is limited to the number of measurements. To overcome the ill-conditioning either the Marquardt method or nth-order regularization is implemented. The e!ects of measurement protocol (number of electrodes and current pattern) and measurement noise are also examined. The global parameterization using Chebyshev polynomials was successful, whereas inverses utilizing the localized parameterization failed to return acceptable images of the free surface. In the majority of cases examined, the Marquardt method for solving iterative least-squares problems gave more accurate solutions than any of the regularization methods within the "rst 100 iterations. Among the regularization methods, "rst-order regularization out-performed both the zeroth- and second-order regularizations. Inverses of data collected with the opposite measurement protocol generally required a fewer number of iterations to reach an acceptable solution than the adjacent measurement protocol, and solving the inverse problem using an overdetermined system was found to b