Abstract
Electromagnetic induction measurements, which are generally used to determine lateral variations of apparent electrical conductivity, can provide quantitative estimates of the subsurface conductivity at different depths. Quantitative inference about the Earth's interior from experimental data is, however, an ill-posed problem. Using the generalised McNeill's theory for the EM38 ground conductivity meter, we generated synthetic apparent conductivity curves (input data vector) simulating measurements at different heights above the soil surface. The electrical conductivity profile (the Earth model) was then estimated solving a least squares problem with Tikhonov regularization optimised with a projected conjugate gradient algorithm. Although the Tikhonov approach improves the conditioning of the resulting linear system, profile reconstruction can be surprisingly far from the desired true one. On the contrary, the projected conjugate gradient provided the best solution without any explicit regularization ( a= 0) of the objective function of the least squares problem. Also, if the initial guess belongs to the image of the system matrix, Im(A), we found that it provides a unique solution in the same subspace Im(A).
Highlights
The goal of collecting geophysical data is to gain meaningful information about a given Earth property
For such a purpose we considered a simple least squares problem for the estimation of the electrical conductivity profile from high-frequency electromagnetic induction
A popular method employed in these applications is the frequency-domain electromagnetic (FEM) induction technique (McNeill, 1980) which uses a ground conductivity meter to measure the apparent electrical conductivity of the subsurface
Summary
The goal of collecting geophysical data is to gain meaningful information about a given Earth property (for example, density, seismic velocity or conductivity of a geologic body). The aim of this work is to illustrate how Tikhonov regularization may sometimes be the cause of misleading results, far from the expected solutions For such a purpose we considered a simple least squares problem for the estimation of the electrical conductivity profile from high-frequency electromagnetic induction. A popular method employed in these applications is the frequency-domain electromagnetic (FEM) induction technique (McNeill, 1980) which uses a ground conductivity meter to measure the apparent electrical conductivity of the subsurface This technique is generally used for detection of lateral changes in the apparent electrical conductivity, it can provide quantitative vertical variations: FEM measurements acquired at different heights above the soil surface can be used to predict the electrical conductivity at different depths
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