Before revision:  The proposed approach is aimed at providing EKI with computational efficiency and robustness for addressing problemswhere the unknown is a heterogeneous physical property with possibly sharp discontinuities arising from the presence of anomalies/defects. In these settings, for EKI to produce accurate estimates of the truth, the unknown needs to be suitably characterised via a parameterisation that often increases the complexity of the identification problem .
After revision:  We motivate our approach using the interpretation of EKI as a Gaussian approximation in the Bayesian tempering setting for inverse problems. We show that our parameter choice controls the symmetrised Kulback-Leibler divergence between consecutive tempering measures. We further motivate our choice using a heuristic statistical discrepancy principle. We test our framework using electrical impedance tomography with the complete electrode model. Parameterisations of the unknown conductivity are employed which enable us to characterise both smooth or a discontinuous (piecewise-constant) fields .