Application of the Variable Projection Scheme for Calibration in Electromagnetic Data Inversion

In this communication, we apply the variable projection method to calibrate measured data for electromagnetic (EM) data inversion. In the calibration process, we need to determine the amplitude and phase of the sources so that the simulated data can match the measured data. These data are difficult to measure due to the casing effect from metallic well pipes. Previous work usually defines the source amplitude and phase as unknowns. They are either computed before or inverted simultaneously during the inversion process. Using the variable projection method, we transform these unknowns into a least-square representation of measured and simulate data. This way we avoid these unknowns in the inversion process. Therefore, this method shows a better computational efficiency. Moreover, it improves the robustness of the inversion especially for gradient methods such as the nonlinear conjugate gradient method. We have incorporated this scheme into the two-and-half-dimensional EM data inversion algorithm and it works effectively in reconstructing the conductivity distribution.