The hybrid extended born approximation and CG-FFHT method for axisymmetric media

The authors develop a hybrid implementation of the extended Born approximation (EBA) with the conjugate-gradient fast Fourier Hankel transform (CG-FFHT) method to improve the efficiency of the numerical solution of borehole induction problems in axisymmetric media. First, they use the FFHT to accelerate the EBA as a nonlinear approximation to induction problems, resulting in an algorithm with O(N log₂ N) arithmetic operations, where N is the number of unknowns in the problem. This accelerated EBA is accurate for most formations encountered in practical applications. Then, for formations with extremely high contrasts, they utilize the accelerated EBA as a partial preconditioner in the CG-FFHT method to solve the problem accurately with few iterations. The seamless combination of these two approaches provides an automatic way toward a general efficient and accurate modeling algorithm for induction measurements in axisymmetric media