4.4 Definition of Data Misfit

In Equation 9a and b, the data misfit is based on an L2 norm (i.e., as the sum of weighted, squared differences between predicted and observed values), which is the first term in the equations. The L2 norm is highly sensitive to outlier data, hence the need to carefully edit datasets and remove data corresponding to electrodes with poor electrical contact or faulty channels. Many inversion packages also support minimization of the L1 data misfit, which has been referred to as robust inversion. The L1 norm is based on absolute differences instead of squared differences and is therefore less sensitive to outliers (Claerbout and Muir, 1973). The L1 norm can also be applied to the regularization term of Equation 9a and b instead of the L2 norm, which tends to produce blocky inversion results with sharp boundaries. Implementation of the L1 norm on either the data misfit or regularization term of the objective function require re-weighting of CD and/or D respectively at each outer iteration of an iterative least-squares solution. This is commonly referred to as Iteratively Reweighted Least Squares inversion (Farquharson and Oldenburg, 1998).


Electrical Imaging for Hydrogeology Copyright © 2022 by Kamini Singha, Timothy C. Johnson, Frederick D. Day Lewis and Lee D. Slater. All Rights Reserved.