Two nonlinear inverse methods for electromagnetic induction measurements

The authors develop two nonlinear inverse methods to reconstruct the conductivity profile from electromagnetic induction (EMI) measurements: the improved two-step inverse method based on the extended Born approximation (EBA) and the combination of the EBA and the contrast source inversion (CSI) method. In the first method, the nonlinear problem is recast as a two-step linear inversion and is solved by using the extended Born approximation. The authors improve this method with the fast Fourier transform (FFT) algorithm and by a conjugate-gradient optimization procedure. For a problem with N unknown pixels and ill measurement data points, the computational time of this inverse EBA procedure is reduced to O(MN) in the first step (i.e., the underdetermined linear problem) and to O(Nlog₂N) in the second step (i.e., the well-determined linear problem) using the newly developed FFT-EBA method. Furthermore, the memory requirement is reduced to O(M_N) (M_T is the number of transmitters). In the second inverse method, they apply the contrast source inversion (CST) method, but use the two-step linear inversion result as the initial solution. By using FFT, the CPU time costs O(C₁MN+C ₂N log₂ N) in each iteration of the CSI procedure. Numerical examples illustrate the effectiveness of these methods, even up to a high contrast of 100:1