Multiscale electrical impedance tomography

[1] Electrical impedance tomography aims to recover the electrical conductivity underground from surface and/or borehole apparent resistivity measurements. This is a highly nonlinear inverse problem, and linearized inverse methods are likely to produce solutions corresponding to local minima of the misfit function to minimize. In the present paper, electrical impedance tomography is addressed through a nonlinear approach, namely, simulated annealing, in order to escape from local minima and to produce conductivity distributions independent of the starting models. Simulated annealing belongs to the Monte Carlo family which needs numerous forward modelings, and particular attention is paid both to the forward numerical solution of the Poisson equation and to the parameterization of the inverse problem, i.e., the way the conductivity distribution is mathematically represented. The 2.5-dimensional forward problem is solved through a multigrid approach which iteratively solves the Poisson equation from large to small scales. In the same spirit, the conductivity model is parameterized in a multiscale way by representing the conductivity distribution with a superimposition of block whose sizes (in suitable units) are integer powers of 2. The multiscale inversion is implemented in a sequential way by first inverting for the conductivity of the coarser blocks and by progressively incorporating finer blocks in the conductivity model. A decision stage based on a sensitivity analysis is performed before incorporating finer blocks in order to optimize the parameterization by not adding unnecessary details into the conductivity model. Both a synthetic example and a field experiment are used to explain the method, and comparisons with a linearized inversion technique are done.